1. Outline
Introduction
Main Theorem
Explicit Formula for the Mean Square of a
Dirichlet L-Function of Prime Power Modulus
Frank Romascavage, III
Thesis under the direction of Professor Djordje Mili´cevi´c
Bryn Mawr College Graduate School of Arts and Sciences
Philadelphia Area Number Theory Seminar
Bryn Mawr College
December 7, 2016
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
3. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Abstract
We seek to derive an explicit formula for the mean square of a
Dirichlet L-function of prime power modulus.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
4. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Abstract
We seek to derive an explicit formula for the mean square of a
Dirichlet L-function of prime power modulus.
Motohashi derived an explicit formula for the mean square (second
power moment) and also the fourth power moment of the Riemann
Zeta Function.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
5. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Abstract
We seek to derive an explicit formula for the mean square of a
Dirichlet L-function of prime power modulus.
Motohashi derived an explicit formula for the mean square (second
power moment) and also the fourth power moment of the Riemann
Zeta Function.
There are many estimates on the power moments for a Dirichlet
L-function, but this work will not include an error term.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
6. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Abstract
We seek to derive an explicit formula for the mean square of a
Dirichlet L-function of prime power modulus.
Motohashi derived an explicit formula for the mean square (second
power moment) and also the fourth power moment of the Riemann
Zeta Function.
There are many estimates on the power moments for a Dirichlet
L-function, but this work will not include an error term.
The values of the L-functions at the central value often have deep
arithmetic significance, but are difficult to study individually.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
7. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Abstract
We seek to derive an explicit formula for the mean square of a
Dirichlet L-function of prime power modulus.
Motohashi derived an explicit formula for the mean square (second
power moment) and also the fourth power moment of the Riemann
Zeta Function.
There are many estimates on the power moments for a Dirichlet
L-function, but this work will not include an error term.
The values of the L-functions at the central value often have deep
arithmetic significance, but are difficult to study individually.
In several important problems in analytic number theory, including
how primes are distributed, estimates such as power moments can
give important information about these central values in families.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
8. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Overview of Review
In this section, we will briefly discuss such topics as the Divisor
Function, Gamma Function, Fourier Transform, Mellin Transform,
Dirichlet Characters, and Dirichlet L-series. We will quickly go over
key concepts of each of these respective areas that will be utilized
throughout the rest of the work.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
10. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Divisor Function
Let d be the divisor function defined by
d(n) =
w|n
1.
Let ∆ be the Delta function given by
∆(x) =
n≤x
d(n) − x(log x + 2γ − 1) −
1
4
,
where in this sum the last term is to be divided by 2 if x ∈ Z.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
12. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Gamma Function
Let Γ : C(−Z ∪ {0}) → C be defined by
Γ(x) =
∞
0
tx−1
e−t
dt, (Re(x) > 0)
Functional equation: Γ(s + 1) = sΓ(s)
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
13. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Gamma Function
Let Γ : C(−Z ∪ {0}) → C be defined by
Γ(x) =
∞
0
tx−1
e−t
dt, (Re(x) > 0)
Functional equation: Γ(s + 1) = sΓ(s)
Euler’s reflection formula: Γ(z)Γ(1 − z) = π
sin(πz)
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
14. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Gamma Function
Let Γ : C(−Z ∪ {0}) → C be defined by
Γ(x) =
∞
0
tx−1
e−t
dt, (Re(x) > 0)
Functional equation: Γ(s + 1) = sΓ(s)
Euler’s reflection formula: Γ(z)Γ(1 − z) = π
sin(πz)
Stirling’s formula:
|Γ(σ + it)| = (2π)1/2|t|σ−1
2 e−π|t|/2 1 + O |t|−1
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
15. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Introduction to Dirichlet Characters
Let q ∈ N. Let (Z/qZ)∗
= {[a]q| gcd(a, q) = 1} and
C∗ = {z ∈ C|z = 0}. A homomorphism χ : (Z/qZ)∗
→ C∗ is
called a Dirichlet character of modulus q.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
16. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Introduction to Dirichlet Characters
Let q ∈ N. Let (Z/qZ)∗
= {[a]q| gcd(a, q) = 1} and
C∗ = {z ∈ C|z = 0}. A homomorphism χ : (Z/qZ)∗
→ C∗ is
called a Dirichlet character of modulus q.
χ(m) =
χ(m(mod q)) if gcd(m, q) = 1
0 if gcd(m, q) > 1
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
17. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Gauss Sum
Let χ be a Dirichlet character of modulus q. Then, we have
that the Gauss sum of χ is given by
τ(χ) =
n(mod q)
χ(n)e2πin/q
.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
18. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Gauss Sum
Let χ be a Dirichlet character of modulus q. Then, we have
that the Gauss sum of χ is given by
τ(χ) =
n(mod q)
χ(n)e2πin/q
.
Note that we have the estimate |τ(χ)| ≤
√
q.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
19. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Orthogonality Relations
Let χ be a Dirichlet character of modulus q and χ0 be the trivial
Dirichlet character of modulus q. Then, we have that
a(mod q)
χ(a) =
ϕ(q) if χ = χ0
0 if χ = χ0.
and
χ(mod q)
χ(m) =
ϕ(q) if m ≡ 1(mod q)
0 otherwise.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
21. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Dirichlet L-series
A Dirichlet L-series is a function of the form
L(s, χ) =
∞
n=1
χ(n)
ns
.
In addition, when Re(s) > 1, we have that L(s, χ) can be
written as an Euler product:
p prime
1 −
χ(p)
ps
−1
.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
22. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Functional Equations for L(s, χ) and ζ(s)
Let χ be a primitive character of modulus q. Let
ξ(s, χ) =
q
π
s/2
Γ
s + aχ
2
L(s, χ),
where aχ = 0 when χ(−1) = 1 and aχ = 1 when
χ(−1) = −1. Then, we have that the following functional
equation holds
L(s, χ) =
τ(χ)
iaχ π1/2
q
π
−s
Γ
1 − s + aχ
2
Γ
s + aχ
2
−1
L(1−s, χ).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
23. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Functional Equations for L(s, χ) and ζ(s)
Let χ be a primitive character of modulus q. Let
ξ(s, χ) =
q
π
s/2
Γ
s + aχ
2
L(s, χ),
where aχ = 0 when χ(−1) = 1 and aχ = 1 when
χ(−1) = −1. Then, we have that the following functional
equation holds
L(s, χ) =
τ(χ)
iaχ π1/2
q
π
−s
Γ
1 − s + aχ
2
Γ
s + aχ
2
−1
L(1−s, χ).
In addition, recall the functional equation for the Riemann
zeta function: ζ(s) = 2sπs−1 sin 1
2sπ Γ(1 − s)ζ(1 − s).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
24. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Dealing with Non-Primitive Characters
Now, we will consider the case of χ not being a primitive character
of modulus q. If χ is a non-primitive character of conductor q |q,
and if χq is the primitive character equivalent to χ then
L(s, χ) = L(s, χq )
p|q,p q
1 −
χq (p)
ps
.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
25. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Fourier Transform
For suitable f : R → C, the Fourier Transform of f is Ff : R → C
defined by
(Ff )(ξ) =
1
2π R
f (t)e−iξt
dt.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
26. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Fourier Transform for Finite Abelian Groups
If G is a finite abelian group, then for f : G → C, the Fourier
Transform (FT) of f , f : G → C (where G is the dual group of G),
is defined by
f (χ) =
g∈G
f (g)χ(g).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
27. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Inverse Fourier Transform
For suitable g : R → C, the Inverse Fourier Transform of g is:
F−1g : R → C defined by
(F−1
g)(t) =
R
g(ξ)eiξt
dξ.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
28. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Fourier Inversion Formula
Let f : G → C. The Fourier Inversion Formula (FIF) is given by
f (x) =
1
|G|
χ∈G
f (χ) χ(x),
where |G| is the order of G.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
29. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Mellin Transform
For suitable f : R+ → C, the Mellin Transform of f is:
Mf : {Re(s) > σ0} → C (where σ0 > 0) defined by
(Mf )(s) =
R+
f (t)ts dt
t
.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
30. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Mellin Transform for Finite Abelian Groups
If G is a finite abelian group, then for f : G → C∗, the Mellin
Transform (MT) of f , denoted by Mf , is given by:
(Mf )(χ) =
y∈G
f (y)χ(y).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
31. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Inverse Mellin Transform
The Inverse Mellin Transform of g is: M−1g : R+ → C defined by
(M−1
g)(t) =
1
2πi Re(s)=σ
g(s)t−s
ds
for any suitable σ (where σ > 0).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
32. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Mellin Inversion Formula
If G is a finite abelian group and f : G → C∗, then the Mellin
Inversion Formula is given by
f (g) =
1
|G|
χ∈G
(Mf )(χ)χ(g).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
34. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Motivation Part 1
Why do we look at moments of L-functions?
Lindel¨of hypothesis for the Riemann zeta function: For any
> 0, ζ 1
2 + it = O(t )
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
35. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Motivation Part 1
Why do we look at moments of L-functions?
Lindel¨of hypothesis for the Riemann zeta function: For any
> 0, ζ 1
2 + it = O(t )
The Lindel¨of hypothesis is true if and only if for all n ∈ N,
1
T
T
0 ζ 1
2 + it
2n
dt = O(T ).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
36. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Motivation Part 2
Generalized Lindel¨of hypothesis (GLH): Suppose L(s, f ) is an
L-function which has a meromorphic continuation to the
entire s-plane and a functional equation relating L(s, f ) and
L(1 − s, f ). Then, for any > 0, L 1
2 + it, f = O(t ).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
37. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Motivation Part 2
Generalized Lindel¨of hypothesis (GLH): Suppose L(s, f ) is an
L-function which has a meromorphic continuation to the
entire s-plane and a functional equation relating L(s, f ) and
L(1 − s, f ). Then, for any > 0, L 1
2 + it, f = O(t ).
We study moments of the form 1
T
T
0 L 1
2 + it, f
2n
dt, for
every n ∈ N to gain a better understanding of the GLH.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
38. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Atkinson’s Formula for the Mean Square
Investigate the asymptotics of
I(T) =
T
0 ζ(1
2 + it)
2
dt = T log T
2π + (2γ − 1)T + E(T)
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
39. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Atkinson’s Formula for the Mean Square
Investigate the asymptotics of
I(T) =
T
0 ζ(1
2 + it)
2
dt = T log T
2π + (2γ − 1)T + E(T)
Let N T and N = T
2π + N
2 − N2
4 + NT
2π
1/2
. Then,
E(T) = 1(T) + 2(T) + O(log2
T), where
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
40. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Atkinson’s Formula for the Mean Square
Investigate the asymptotics of
I(T) =
T
0 ζ(1
2 + it)
2
dt = T log T
2π + (2γ − 1)T + E(T)
Let N T and N = T
2π + N
2 − N2
4 + NT
2π
1/2
. Then,
E(T) = 1(T) + 2(T) + O(log2
T), where
1(T) =
√
2 T
2π
1/4
n≤N(−1)nd(n)n−3/4e(T, n)
cos(f (T, n)),
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
41. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Atkinson’s Formula for the Mean Square
Investigate the asymptotics of
I(T) =
T
0 ζ(1
2 + it)
2
dt = T log T
2π + (2γ − 1)T + E(T)
Let N T and N = T
2π + N
2 − N2
4 + NT
2π
1/2
. Then,
E(T) = 1(T) + 2(T) + O(log2
T), where
1(T) =
√
2 T
2π
1/4
n≤N(−1)nd(n)n−3/4e(T, n)
cos(f (T, n)),
2(T) = −2 n≤N d(n)n−1/2 log T
2πn
−1
cos(g(T, n)),
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
42. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Atkinson’s Formula for the Mean Square
Investigate the asymptotics of
I(T) =
T
0 ζ(1
2 + it)
2
dt = T log T
2π + (2γ − 1)T + E(T)
Let N T and N = T
2π + N
2 − N2
4 + NT
2π
1/2
. Then,
E(T) = 1(T) + 2(T) + O(log2
T), where
1(T) =
√
2 T
2π
1/4
n≤N(−1)nd(n)n−3/4e(T, n)
cos(f (T, n)),
2(T) = −2 n≤N d(n)n−1/2 log T
2πn
−1
cos(g(T, n)),
e(T, n) = 1 + O n
T , f (T, n) = 4π πn
2T − π
4 +
O n3/2T−1/2 , g(T, n) = T log T
2πn − T + π
4
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
43. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
What Motohashi Accomplished Part 1
Motohashi was able to obtain an exact formula for
L1(g) =
∞
−∞ ζ 1
2 + it
2
g(t) dt when g is a regular
function that satisfies g(r) = O (|r| + 1)−A
for |Im(r)| ≤ A
where A is a large positive constant.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
44. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
What Motohashi Accomplished Part 2
Motohashi’s formula:
L1(g) =
∞
−∞
Re
Γ
Γ
1
2
+ it + 2cE − log(2π) g(t) dt + 2π
Re g
1
2
i + 4
∞
n=1
d(n)
∞
0
(y(y + 1))−1/2
gc log 1 +
1
y
cos (2πny) dy,
where cE is Euler’s constant and
gc(x) =
∞
−∞
g(t) cos(xt) dt.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
45. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
What Motohashi Accomplished Part 3
Motohashi achieved this result in several steps.
First, examine J(u, v; g) =
∞
−∞ ζ(u + it)ζ(v − it)g(t) dt in
order to get a better understanding of L1(g)
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
46. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
What Motohashi Accomplished Part 3
Motohashi achieved this result in several steps.
First, examine J(u, v; g) =
∞
−∞ ζ(u + it)ζ(v − it)g(t) dt in
order to get a better understanding of L1(g)
Second, divide J(u, v; g) into diagonal and off-diagonal sums
in the area of absolute convergence, i.e.
J(u, v; g) =
m=n
+
m<n
+
m>n
m−u
n−v
g∗
log
n
m
= ζ(u + v)g∗
(0) + J1(u, v; g) + J1(v, u; g),
where g∗ is the Fourier Transform of g.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
47. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
What Motohashi Accomplished Part 4
Third, continue J1 meromorphically to the domain
|u|, |v| < B = cA where c is a small constant and A is a large
constant.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
48. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
What Motohashi Accomplished Part 4
Third, continue J1 meromorphically to the domain
|u|, |v| < B = cA where c is a small constant and A is a large
constant.
This occurs by using the Mellin Transform, Stirling’s Formula,
the Inverse Mellin Transform, and then moving the path of
integration by contour integration.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
49. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
What Motohashi Accomplished Part 4
Third, continue J1 meromorphically to the domain
|u|, |v| < B = cA where c is a small constant and A is a large
constant.
This occurs by using the Mellin Transform, Stirling’s Formula,
the Inverse Mellin Transform, and then moving the path of
integration by contour integration.
Next, the functional equation of the Riemann zeta function is
utilized.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
50. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
What Motohashi Accomplished Part 4
Third, continue J1 meromorphically to the domain
|u|, |v| < B = cA where c is a small constant and A is a large
constant.
This occurs by using the Mellin Transform, Stirling’s Formula,
the Inverse Mellin Transform, and then moving the path of
integration by contour integration.
Next, the functional equation of the Riemann zeta function is
utilized.
Since the region |u|, |v| < B = cA is symmetric, we can take
u = v = α.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
51. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
What Motohashi Accomplished Part 5
Fourth, the path of integration is moved again by contour
integration.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
52. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
What Motohashi Accomplished Part 5
Fourth, the path of integration is moved again by contour
integration.
Fifth, α = 1
2 is examined. Singularities are canceled out as α
approaches 1
2, the remaining integral is transformed, and then
the desired result is obtained.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
54. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Analogy
We want to find an analogous formula for
L1(g) =
∞
−∞
ζ
1
2
+ it
2
g(t) dt
Integrating over all t ∈ R is analogous to summing over all
characters χ(mod pβ).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
55. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Analogy
We want to find an analogous formula for
L1(g) =
∞
−∞
ζ
1
2
+ it
2
g(t) dt
Integrating over all t ∈ R is analogous to summing over all
characters χ(mod pβ).
By comparing ζ(u + it) with L(u, χ), we note that χ(n) plays
the role of n−it in Motohashi: ζ(u + it) = ∞
n=1
1
nunit and
L(u, χ) = ∞
n=1
χ(n)
nu .
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
56. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Results Part 1
P. X. Gallagher (1975): For q ≥ 2 and t ∈ R,
χ(mod q)
L
1
2
+ it, χ
2
= O ((q + |t|) log q (|t| + 2))
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
57. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Results Part 2
R. Balasubramanian (1980): For q ≥ 2 and t ≥ 3,
χ(mod q)
L
1
2
+ it, χ
2
=
ϕ2(q)
q
log qt + O q (log log q)2
+
O te10
√
log q
+ O q
1
2 t
2
3 e10
√
log q
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
58. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Results Part 3
D. R. Heath-Brown (1981):
χ(mod q)
L
1
2
, χ
2
=
ϕ(q)
q
k|q
µ
q
k
T(k),
where
T(k) = k log
k
8π
+ γ + 2ζ2 1
2
k
1
2 +
2N−1
n=0
cnk−n
2 + O k−N
for any integer N ≥ 1, numerical constants cn, and Euler’s
constant γ.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
59. Outline
Introduction
Main Theorem
Abstract
Review
Motivation
Atkinson’s Formula for the Mean Square
Motohashi’s Formula
Analogy
Results
Results Part 4
Zhang Wenpeng (1989 - 1991):
χ(mod q)
L
1
2
+ it, χ
2
=
ϕ2(q)
q
log
qt
2π
+ 2γ +
p|q
log p
p − 1
+
O qt−1/12
+ O t5/6
+ q1/2
t5/12
exp
2 log(qt)
log log(qt)
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
60. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1 of Statement
Let p be a prime, β ∈ N, ν ∈ N such that 0 ≤ ν < β, and g be an
arbitrary real-valued weight function on (Z/pβZ)∗.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
61. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1 of Statement
Let p be a prime, β ∈ N, ν ∈ N such that 0 ≤ ν < β, and g be an
arbitrary real-valued weight function on (Z/pβZ)∗.
Let F1(g) =
χ∈(Z/pβ Z)∗ L 1
2
, χ
2
g (χ) where g is a weight function.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
62. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1 of Statement
Let p be a prime, β ∈ N, ν ∈ N such that 0 ≤ ν < β, and g be an
arbitrary real-valued weight function on (Z/pβZ)∗.
Let F1(g) =
χ∈(Z/pβ Z)∗ L 1
2
, χ
2
g (χ) where g is a weight function.
Let C = 2γ +
Γ (1
2 )
Γ(1
2 )
, where Γ is the Gamma function and γ is the
Euler-Mascheroni constant.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
63. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1 of Statement
Let p be a prime, β ∈ N, ν ∈ N such that 0 ≤ ν < β, and g be an
arbitrary real-valued weight function on (Z/pβZ)∗.
Let F1(g) =
χ∈(Z/pβ Z)∗ L 1
2
, χ
2
g (χ) where g is a weight function.
Let C = 2γ +
Γ (1
2 )
Γ(1
2 )
, where Γ is the Gamma function and γ is the
Euler-Mascheroni constant.
Let τ be the Gauss sum. Let d be the divisor function and ∆ be the
Delta function.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
64. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1 of Statement
Let p be a prime, β ∈ N, ν ∈ N such that 0 ≤ ν < β, and g be an
arbitrary real-valued weight function on (Z/pβZ)∗.
Let F1(g) =
χ∈(Z/pβ Z)∗ L 1
2
, χ
2
g (χ) where g is a weight function.
Let C = 2γ +
Γ (1
2 )
Γ(1
2 )
, where Γ is the Gamma function and γ is the
Euler-Mascheroni constant.
Let τ be the Gauss sum. Let d be the divisor function and ∆ be the
Delta function.
Let f : C × R → C be defined by f (u, x) = 2
∞
0
y−u
(y + 1)u−1
cos(2π xy).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
65. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1 of Statement
Let p be a prime, β ∈ N, ν ∈ N such that 0 ≤ ν < β, and g be an
arbitrary real-valued weight function on (Z/pβZ)∗.
Let F1(g) =
χ∈(Z/pβ Z)∗ L 1
2
, χ
2
g (χ) where g is a weight function.
Let C = 2γ +
Γ (1
2 )
Γ(1
2 )
, where Γ is the Gamma function and γ is the
Euler-Mascheroni constant.
Let τ be the Gauss sum. Let d be the divisor function and ∆ be the
Delta function.
Let f : C × R → C be defined by f (u, x) = 2
∞
0
y−u
(y + 1)u−1
cos(2π xy).
Let Tg (z) = 1
ϕ(q) ψ∈(Z/pβ Z)∗
g(ψ)ψ(z) and ˜gβ−ν (y) = y(mod pβ−ν )
Tg (1 + pν
y)χ(y)
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
66. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 2 of Statement
Let W1 be defined by
W1(b) = π−1/2
(2)
2bπ
pβ−ν
−s
Γ2
(s)Γ
1
2
− s e−isπ/2
ds +
(2)
2bπ
pβ−ν
−s
Γ2
(s)Γ
1
2
− s eisπ/2
ds
and W2 be defined by
W2(b) = π1/2
i
(2)
2bπ
pβ−ν
−s
Γ2
(s)Γ
1
2
− s e−isπ/2
ds −
(2)
2bπ
pβ−ν
−s
Γ2
(s)Γ
1
2
− s eisπ/2
ds .
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
67. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 3 of Statement
Then, F1(g) = A1(g) + A2(g) + A3(g) + A4(g), where
A1(g) := pβ
1 −
1
p
2
χ∈(Z/pβZ)∗
g (χ)
log p
1 − p
+ log
pβ
2π
+ C+
4p−β
1 −
1
p
−1
=pβ,pβ−1
µ
pβ
f
1
2
, 1 − f
1
2
,
3
2
∆
3
2
+
(π )−1
∞
0
y3
(y + 1)
−1/2
sin (3π y)
1
2
− log
3
2
+ 2γ (y + 1)
−
∞
3/2
∆(t)
∂f 1
2, t
∂t
dt ,
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
68. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 4 of Statement
A2(g) :=
0≤ν<β
pν p − 1
p
Re(˜gβ−ν(χ0)) log p
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
69. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 5 of Statement
A3(g) := −
2
π
0≤ν<β
pν
∞
f =1
d(f )f −1/2
j=−1,−2
f j
Γ
1
2
− j
(−1)−j
Γ(1 − j)
p2νj
pβ
π
− 1
2 −j
∗
χ(mod pβ−ν )
Γ
1
2 −j+aχ
2
Γ
1
2 +j+aχ
2
Re i−aχ
τ(χ)˜gβ−ν(χ)χ(f ) +
β−ν
γ=1 χpγ ≡χ
π−1/2
Γ
aχpγ −j+ 1
2
2
Γ
aχpγ + 1
2 +j
2
Re i−aχpγ
τ(χpγ )˜gβ−ν(χpγ )χpγ (f ) ,
and
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
70. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 6 of Statement
A4(g) :=
1
πpβ
0≤ν<β
p2ν
∞
b=1
d(b) Im W1(b)
∗
χ(mod pβ−ν )
χ even
τ(χ)˜gβ−ν(χ)
χ(b) +
β−ν
γ=1 χpγ ≡χ
χpγ even
τ(χpγ )˜gβ−ν(χpγ )χpγ (b) + Re W2(b)
∗
χ(mod pβ−ν )
χ odd
τ(χ)˜gβ−ν(χ)χ(b) +
χpγ ≡χ
χpγ odd
τ(χpγ )˜gβ−ν(χpγ )χpγ (b) .
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
71. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Introducing F(u, v; g)
When dealing with F1(g) we can consider
F(u, v; g) =
χ∈(Z/pβZ)
∗
L(u, χ)L(v, χ)g (χ)
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
72. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Expansion of F(u, v; g)
Since we are starting with Re(u), Re(v) > 1, we have that
F(u, v; g) =
χ∈(Z/pβZ)
∗
∞
m=1
χ(m)
mu
∞
n=1
χ(n)
nv
g (χ)
=
∞
m=1
∞
n=1
m−u
n−v
χ∈(Z/pβZ)
∗
χ
m
n
g (χ)
= ϕ(pβ
)
∞
m=1
∞
n=1
m−u
n−v
Tg
n
m
.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
73. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Orthogonality, Diagonal, Near-diagonal, and Off-diagonal
Sums
By orthogonality and breaking up F(u, v; g) into diagonal, near-diagonal, and
off-diagonal sums, we get that F(u, v; g) is equal to
ϕ(pβ
)
m=n
(m,pβ
)=(n,pβ
)=1
+
m<n
m≡n(mod pβ
)
(m,pβ
)=(n,pβ
)=1
+
m>n
m≡n(mod pβ
)
(m,pβ
)=(n,pβ
)=1
+
m<n
m≡n(mod pβ
)
(m,pβ
)=(n,pβ
)=1
+
m>n
m≡n(mod pβ
)
(m,pβ
)=(n,pβ
)=1
m−u
n−v
Tg
n
m
. (Call each of these sums F1(u, v; g)(diagonal)
F2(u, v; g) (near-diagonal), F3(u, v; g)(near-diagonal), F4(u, v; g) (off-diagonal) and
F5(u, v; g)(off-diagonal), respectively.)
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
74. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Big Picture
The first three terms, diagonal and near-diagonal sums,
contribute to the main term, which is A1(g).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
75. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Big Picture
The first three terms, diagonal and near-diagonal sums,
contribute to the main term, which is A1(g).
Contour shifting and analytic continuation will be used.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
76. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Big Picture
The first three terms, diagonal and near-diagonal sums,
contribute to the main term, which is A1(g).
Contour shifting and analytic continuation will be used.
The off-diagonal sums will contribute to the sum terms, A3(g)
and A4(g), and A2(g) as well.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
77. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Big Picture
The first three terms, diagonal and near-diagonal sums,
contribute to the main term, which is A1(g).
Contour shifting and analytic continuation will be used.
The off-diagonal sums will contribute to the sum terms, A3(g)
and A4(g), and A2(g) as well.
Mellin transform and Mellin Inversion Formula are used for the
last two.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
78. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Evaluation of F1(u, v; g)
Notice that
F1(u, v; g) = L(u + v, χ0)
χ∈(Z/pβZ)
∗
g(χ)
by appealing directly from F(u, v; g).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
79. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Expanding F2(u, v; g)
By appealing directly from F(u, v; g),
F2(u, v; g) =
χ∈(Z/pβZ)
∗
g(χ)
m
(m,pβ)=1
∞
j=1
m−u
(m + jpβ
)−v
.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
80. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Relating F3(u, v; g) to F2(u, v; g)
Note that by utilizing the definitions of F2(u, v; g) and F3(u, v; g)
and complex conjugation we get that
F2(v, u; g) = F3(u, v; g).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
81. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Introducing fpβ (u, v) and simplifying
F1(u, v; g) + F2(u, v; g) + F3(u, v; g)
Let
fpβ (u, v) =
m
(m,pβ)=1
∞
j=1
m−u
(m + jpβ
)−v
.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
82. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Introducing fpβ (u, v) and simplifying
F1(u, v; g) + F2(u, v; g) + F3(u, v; g)
Let
fpβ (u, v) =
m
(m,pβ)=1
∞
j=1
m−u
(m + jpβ
)−v
.
Then, we have that
F1(u, v; g) + F2(u, v; g) + F3(u, v; g) = L(u + v, χ0)+
fpβ (u, v) + fpβ (v, u)
χ∈(Z/pβZ)
∗
g (χ) .
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
83. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Taking fpβ (u, v) from R1 = Re(u) > 1, Re(v) > 1
to R2 = Re(u) < −1, Re(u + v) > 2
In the first step, we write fpβ (u, v) as double sum when
Re(u) > 1 and Re(v) > 1:
fpβ (u, v) =
|pβ
µ( )
∞
r=1
∞
j=1
( r)−u
( r + jpβ
)−v
,
where µ is the M¨obius function.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
84. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Taking fpβ (u, v) from R1 = Re(u) > 1, Re(v) > 1
to R2 = Re(u) < −1, Re(u + v) > 2
In the first step, we write fpβ (u, v) as double sum when
Re(u) > 1 and Re(v) > 1:
fpβ (u, v) =
|pβ
µ( )
∞
r=1
∞
j=1
( r)−u
( r + jpβ
)−v
,
where µ is the M¨obius function.
In the second step, we rearrange the double sum and note
that we can do this when Re(v) > 1 and Re(u + v) > 2.
∞
r=1
∞
j=1
( r)−u
( r + jpβ
)−v
=
∞
j=1
∞
r=1
( r)−u
( r + jpβ
)−v
.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
85. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 2: Taking fpβ (u, v) from R1 to R2
In the third step, we will use Poisson Summation to note that
when Re(u) < −1 and Re(u + v) > 2 that
∞
j=1
∞
r=1
( r)−u
( r + jpβ
)−v
=
∞
0
( x)−u
( x + jpβ
)−v
dx+
2
∞
r=1
∞
0
( x)−u
( x + jpβ
)−v
cos(2πrx) dx.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
86. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 3: Taking fpβ (u, v) from R1 to R2
In the fourth step, we will substitute z = x, simplify, and
then substitute y = z
jpβ to obtain when Re(u) < −1 and
Re(u + v) > 2 that
∞
0
( x)−u
( x + jpβ
)−v
dx + 2
∞
r=1
∞
0
( x)−u
( x + jpβ
)−v
cos(2πrx) dx = −1
(jpβ
)1−u−v
∞
0
y−u
(y + 1)−v
dy + 2
∞
r=1
∞
0
y−u
(y + 1)−v
cos 2πr
jpβy
dy .
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
87. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Defining Gpβ (u, v)
Define
Gpβ (u, v) := 2(pβ
)1−u−v
|pβ
µ( ) −1
∞
j=1
j1−u−v
∞
r=1
∞
0
y−u
(1 + y)−v
cos 2πr
pβ
jy dy.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
88. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Determining fpβ (u, v) + fpβ (v, u)
Then, when Re(u) < −1 and Re(u + v) > 2, we have that
fpβ (u, v) + fpβ (v, u) = ϕ(pβ
)(pβ
)−u−v
ζ(u + v − 1)Γ(u + v − 1)
Γ(1 − u)
Γ(v)
+
Γ(1 − v)
Γ(u)
+ Gpβ (u, v) + Gpβ (v, u).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
89. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Determining fpβ (u, v) + fpβ (v, u)
Then, when Re(u) < −1 and Re(u + v) > 2, we have that
fpβ (u, v) + fpβ (v, u) = ϕ(pβ
)(pβ
)−u−v
ζ(u + v − 1)Γ(u + v − 1)
Γ(1 − u)
Γ(v)
+
Γ(1 − v)
Γ(u)
+ Gpβ (u, v) + Gpβ (v, u).
This is achieved by: substitution from previous evaluations for
fpβ (u, v), substituting jpβy for x, and utilizing that
|pβ µ( ) −1 = ϕ(pβ)
pβ , ∞
j=1 j−1−u−v = ζ(u + v − 1), and
∞
0 y−u(y + 1)−v dy = Γ(u + v − 1)Γ(1−u)
Γ(v) .
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
90. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Analytic Continuation of F1 + F2 + F3 from R1 to
R2
Notice that fpβ (u, v) − Gpβ (u, v) = ϕ(pβ) pβ −u−v
ζ(u + v − 1)Γ(u + v − 1)Γ(1−u)
Γ(v) .
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
91. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Analytic Continuation of F1 + F2 + F3 from R1 to
R2
Notice that fpβ (u, v) − Gpβ (u, v) = ϕ(pβ) pβ −u−v
ζ(u + v − 1)Γ(u + v − 1)Γ(1−u)
Γ(v) .
Note that the right-hand side does in fact provide an analytic
continuation of fpβ (u, v) − Gpβ (u, v) to a meromorphic
function of u and v.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
92. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Analytic Continuation of F1 + F2 + F3 from R1 to
R2
Notice that fpβ (u, v) − Gpβ (u, v) = ϕ(pβ) pβ −u−v
ζ(u + v − 1)Γ(u + v − 1)Γ(1−u)
Γ(v) .
Note that the right-hand side does in fact provide an analytic
continuation of fpβ (u, v) − Gpβ (u, v) to a meromorphic
function of u and v.
So, we have that fpβ (u, v) − Gpβ (u, v) + fpβ (v, u) − Gpβ (v, u)
= ϕ(pβ)(pβ)−u−v ζ(u + v − 1)Γ(u + v − 1) Γ(1−u)
Γ(v) + Γ(1−v)
Γ(u)
is meromorphic.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
93. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 2: Analytic Continuation of F1 + F2 + F3 from R1 to
R2
This provides an expression for fpβ (u, v) + fpβ (v, u) if this
sum can be continued analytically, but from F1(u, v; g)+
F2(u, v; g)+F3(u, v; g) = L(u+v, χ0)+fpβ (u, v)+fpβ (v, u)
χ∈(Z/pβZ)
∗ g (χ) it definitely can be.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
94. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Show that Gpβ (u, v) is analytic in a region
containing Re(u) = 1
2, Re(v) = 1
2
Recall that
Gpβ (u, v) = 2(pβ
)1−u−v
|pβ
µ( ) −1
∞
j=1
j1−u−v
∞
r=1
∞
0
y−u
(1 + y)−v
cos 2πr
pβ
jy dy.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
95. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Show that Gpβ (u, v) is analytic in a region
containing Re(u) = 1
2, Re(v) = 1
2
Recall that
Gpβ (u, v) = 2(pβ
)1−u−v
|pβ
µ( ) −1
∞
j=1
j1−u−v
∞
r=1
∞
0
y−u
(1 + y)−v
cos 2πr
pβ
jy dy.
First, we examine the convergence of 2
∞
0
y−u
(y + 1)−v
cos 2πr pβ
jy dy.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
96. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Show that Gpβ (u, v) is analytic in a region
containing Re(u) = 1
2, Re(v) = 1
2
Recall that
Gpβ (u, v) = 2(pβ
)1−u−v
|pβ
µ( ) −1
∞
j=1
j1−u−v
∞
r=1
∞
0
y−u
(1 + y)−v
cos 2πr
pβ
jy dy.
First, we examine the convergence of 2
∞
0
y−u
(y + 1)−v
cos 2πr pβ
jy dy.
This will become 2
∞
0
y−u
(y + 1)−v
cos (2πny) dy by letting n = rpβ
j
(pβ
/ is an integer due to summing over all that divide pβ
).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
97. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 2: Show that Gpβ (u, v) is analytic in a region
containing Re(u) = 1
2, Re(v) = 1
2
Contour integration and estimation can be used to show that
2
∞
0
y−u
(y + 1)−v
cos (2πny) dy = nu−1
i∞
0
y−u y
n
+ 1
−v
e2πiy
dy +
−i∞
0
y−u y
n
+ 1
−v
e−2πiy
dy
|n|Re(u)−1
|1 − Re(u)|
.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
98. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 3: Show that Gpβ (u, v) is analytic in a region
containing Re(u) = 1
2, Re(v) = 1
2
We use the previous estimate to establish that when
Re(u) < 0, Re(v) > 1 and Re(u + v) > 0, Gpβ (u, v) converges
absolutely.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
99. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 3: Show that Gpβ (u, v) is analytic in a region
containing Re(u) = 1
2, Re(v) = 1
2
We use the previous estimate to establish that when
Re(u) < 0, Re(v) > 1 and Re(u + v) > 0, Gpβ (u, v) converges
absolutely.
This is achieved by noting that Gpβ (u, v)
2(pβ)1−u−v
|pβ µ( ) −1 ∞
j=1 j1−u−v ∞
n=1
|n|Re(u)−1
|1−Re(u)| .
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
100. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 4: Show that Gpβ (u, v) is analytic in a region
containing Re(u) = 1
2, Re(v) = 1
2
Now, we will provide a continuation for Gpβ (u, v) from
Re(u) < 0, Re(v) > 1 and Re(u + v) > 0 to Re(u + v)
= 1 + δ, where |δ| < 1
2.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
101. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 4: Show that Gpβ (u, v) is analytic in a region
containing Re(u) = 1
2, Re(v) = 1
2
Now, we will provide a continuation for Gpβ (u, v) from
Re(u) < 0, Re(v) > 1 and Re(u + v) > 0 to Re(u + v)
= 1 + δ, where |δ| < 1
2.
We will soon see that the divisor function appears in analogy
with Motohashi.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
102. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 4: Show that Gpβ (u, v) is analytic in a region
containing Re(u) = 1
2, Re(v) = 1
2
Now, we will provide a continuation for Gpβ (u, v) from
Re(u) < 0, Re(v) > 1 and Re(u + v) > 0 to Re(u + v)
= 1 + δ, where |δ| < 1
2.
We will soon see that the divisor function appears in analogy
with Motohashi.
The divisor function will be multiplied by a smooth weight.
The smooth weight is dealt with by summation by parts.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
103. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 4: Show that Gpβ (u, v) is analytic in a region
containing Re(u) = 1
2, Re(v) = 1
2
Now, we will provide a continuation for Gpβ (u, v) from
Re(u) < 0, Re(v) > 1 and Re(u + v) > 0 to Re(u + v)
= 1 + δ, where |δ| < 1
2.
We will soon see that the divisor function appears in analogy
with Motohashi.
The divisor function will be multiplied by a smooth weight.
The smooth weight is dealt with by summation by parts.
We use results on the divisor problem and advances in it are
why we ultimately end up succeeding.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
104. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 4: Show that Gpβ (u, v) is analytic in a region
containing Re(u) = 1
2, Re(v) = 1
2
Now, we will provide a continuation for Gpβ (u, v) from
Re(u) < 0, Re(v) > 1 and Re(u + v) > 0 to Re(u + v)
= 1 + δ, where |δ| < 1
2.
We will soon see that the divisor function appears in analogy
with Motohashi.
The divisor function will be multiplied by a smooth weight.
The smooth weight is dealt with by summation by parts.
We use results on the divisor problem and advances in it are
why we ultimately end up succeeding.
We use the estimate: ∆(t) t1/3 log t.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
105. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 5: Show that Gpβ (u, v) is analytic in a region
containing Re(u) = 1
2, Re(v) = 1
2
We can write
Gpβ (u, 1 − u) = Gpβ,1(u) − Gpβ,2(u) + Gpβ,3(u) − Gpβ,4(u)
where
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
106. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 5: Show that Gpβ (u, v) is analytic in a region
containing Re(u) = 1
2, Re(v) = 1
2
We can write
Gpβ (u, 1 − u) = Gpβ,1(u) − Gpβ,2(u) + Gpβ,3(u) − Gpβ,4(u)
where
Gpβ,1(u) = 2p−β
|pβ µ pβ
m≤M d(m)f (u, m),
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
107. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 5: Show that Gpβ (u, v) is analytic in a region
containing Re(u) = 1
2, Re(v) = 1
2
We can write
Gpβ (u, 1 − u) = Gpβ,1(u) − Gpβ,2(u) + Gpβ,3(u) − Gpβ,4(u)
where
Gpβ,1(u) = 2p−β
|pβ µ pβ
m≤M d(m)f (u, m),
Gpβ,2(u) = 2p−β
|pβ µ pβ
f u, M + 1
2 ∆ M + 1
2 ,
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
108. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 5: Show that Gpβ (u, v) is analytic in a region
containing Re(u) = 1
2, Re(v) = 1
2
We can write
Gpβ (u, 1 − u) = Gpβ,1(u) − Gpβ,2(u) + Gpβ,3(u) − Gpβ,4(u)
where
Gpβ,1(u) = 2p−β
|pβ µ pβ
m≤M d(m)f (u, m),
Gpβ,2(u) = 2p−β
|pβ µ pβ
f u, M + 1
2 ∆ M + 1
2 ,
Gpβ,3(u) = 2p−β
|pβ µ pβ ∞
M+1
2
f (u, t)(log t + 2γ) dt,
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
109. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 5: Show that Gpβ (u, v) is analytic in a region
containing Re(u) = 1
2, Re(v) = 1
2
We can write
Gpβ (u, 1 − u) = Gpβ,1(u) − Gpβ,2(u) + Gpβ,3(u) − Gpβ,4(u)
where
Gpβ,1(u) = 2p−β
|pβ µ pβ
m≤M d(m)f (u, m),
Gpβ,2(u) = 2p−β
|pβ µ pβ
f u, M + 1
2 ∆ M + 1
2 ,
Gpβ,3(u) = 2p−β
|pβ µ pβ ∞
M+1
2
f (u, t)(log t + 2γ) dt,
Gpβ,4(u) = 2p−β
|pβ µ pβ ∞
M+1
2
∆(t)∂f (u,t)
∂t dt, M ∈ N,
and f (u, x) = 2
∞
0 yu(y + 1)u−1 cos(2π xy) dy.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
110. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 6: Show that Gpβ (u, v) is analytic in a region
containing Re(u) = 1
2, Re(v) = 1
2
Each of the four previous summands analytically continue Gpβ (u, 1 − u) from
Re(u) > 0 to Re(u) < 2
3
and Re(v) = Re(1 − u) > 1
3
which contains Re(u) = 1
2
,
Re(v) = 1
2
(due to the 1
3
’s coming from the bounds from the divisor problem)
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
111. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 6: Show that Gpβ (u, v) is analytic in a region
containing Re(u) = 1
2, Re(v) = 1
2
Each of the four previous summands analytically continue Gpβ (u, 1 − u) from
Re(u) > 0 to Re(u) < 2
3
and Re(v) = Re(1 − u) > 1
3
which contains Re(u) = 1
2
,
Re(v) = 1
2
(due to the 1
3
’s coming from the bounds from the divisor problem)
Thus, by substitution, we have that
Gpβ (u, 1 − u) = 2p−β
|pβ
µ
pβ
f (u, 1) − f u,
3
2
∆
3
2
− log
3
2
+
2γ (π )−1
∞
0
y−u−1
(y + 1)u−1
sin (3π y) dy + (π u)−1
∞
0
y−u−1
(y + 1)u
sin (3π y) dy −
∞
3
2
∆(t)
∂f (u, t)
∂t
dt .
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
112. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Taking F1 + F2 + F3 from R2 = Re(u) < −1, Re(u + v) > 2
to R3 = Re(u + v) = 1 + δ where |δ| < 1
2
Write 1 + δ in place of u + v
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
113. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Taking F1 + F2 + F3 from R2 = Re(u) < −1, Re(u + v) > 2
to R3 = Re(u + v) = 1 + δ where |δ| < 1
2
Write 1 + δ in place of u + v
Use the functional equation for the Riemann zeta function
and Euler’s reflection formula
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
114. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Taking F1 + F2 + F3 from R2 = Re(u) < −1, Re(u + v) > 2
to R3 = Re(u + v) = 1 + δ where |δ| < 1
2
Write 1 + δ in place of u + v
Use the functional equation for the Riemann zeta function
and Euler’s reflection formula
Writing the Riemann zeta function with Big O notation:
ζ(1 + δ) = 1
δ + γ + O(|δ|) (permissible due to ζ(1 + δ) being
near 1 + δ = 1)
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
115. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Taking F1 + F2 + F3 from R2 = Re(u) < −1, Re(u + v) > 2
to R3 = Re(u + v) = 1 + δ where |δ| < 1
2
Write 1 + δ in place of u + v
Use the functional equation for the Riemann zeta function
and Euler’s reflection formula
Writing the Riemann zeta function with Big O notation:
ζ(1 + δ) = 1
δ + γ + O(|δ|) (permissible due to ζ(1 + δ) being
near 1 + δ = 1)
Taylor expansions of each of the terms and expanding the
products
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
116. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
When δ → 0...
Note that as δ → 0, F1(u, v; g) + F2(u, v; g) + F3(u, v; g) in
Re(u + v) = 1 + δ, where |δ| < 1
2, approaches
χ∈(Z/pβZ)∗
g (χ)
ϕ(pβ)
pβ
p|pβ
log p
1 − p
+ 2γ + log
pβ
2π
+
1
2
Γ (1 − u)
Γ(1 − u)
+
1
2
Γ (u)
Γ(u)
+ Gpβ (u, 1 − u) + Gpβ (1 − u, u) .
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
117. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Evaluating F1
1
2, 1
2; g + F2
1
2, 1
2; g + F3
1
2, 1
2; g
Thus, we have that
(F1 + F2 + F3)
1
2
,
1
2
; g = 1 −
1
p
χ∈(Z/pβ Z)∗
g (χ)
log p
1 − p
+ log
pβ
2π
+ C + 4p−β
1 −
1
p
−1
=pβ ,pβ−1
µ
pβ
f
1
2
, 1 − f
1
2
,
3
2
∆
3
2
+ (π )−1
∞
0
y3
(y + 1)
−1/2
sin(3π y)
1
2
− log
3
2
+ 2γ (y + 1) dy −
∞
3/2
∆(t)
∂f 1
2
, t
∂t
dt ,
which is precisely A1(g) (depends only on
χ∈(Z/pβ Z)∗
g (χ), not on individual values
of g(χ)).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
118. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Reviewing F4 and F5 and Relating Them to Each Other
When Re(u) > 1 and Re(v) > 1, recall that
F4(u, v; g) := ϕ(pβ
)
m<n
m≡n(mod pβ)
(m,pβ)=(n,pβ)=1
m−u
n−v
Tg
n
m
and
F5(u, v; g) := ϕ(pβ
)
m>n
m≡n(mod pβ)
(m,pβ)=(n,pβ)=1
m−u
n−v
Tg
n
m
.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
119. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Reviewing F4 and F5 and Relating Them to Each Other
When Re(u) > 1 and Re(v) > 1, recall that
F4(u, v; g) := ϕ(pβ
)
m<n
m≡n(mod pβ)
(m,pβ)=(n,pβ)=1
m−u
n−v
Tg
n
m
and
F5(u, v; g) := ϕ(pβ
)
m>n
m≡n(mod pβ)
(m,pβ)=(n,pβ)=1
m−u
n−v
Tg
n
m
.
As from before, we have that F4(v, u; g) = F5(u, v; g).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
120. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Taking F4 from R1 to R4 = Re(u + v) > 3
First, we will further simplify F4 to obtain
ϕ(pβ
)
m,n>0
m≡n+m(mod pβ)
(m,pβ)=(n+m,pβ)=1
m−u−v n
m
+ 1
−v
Tg
n
m
+ 1 .
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
121. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Taking F4 from R1 to R4 = Re(u + v) > 3
First, we will further simplify F4 to obtain
ϕ(pβ
)
m,n>0
m≡n+m(mod pβ)
(m,pβ)=(n+m,pβ)=1
m−u−v n
m
+ 1
−v
Tg
n
m
+ 1 .
Let n = pνn1 where n1 ∈ N, (n1, p) = 1, and 0 ≤ ν < β.
Note that by the Inverse Mellin Transform,
n
m + 1
−v
= 1
2πi (σ) F(s, v) pν n1
m
−s
ds, where
F(s, v) = R+ (x + 1)−v xs−1 dx and Re(σ) > 0.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
122. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 2: Taking F4 from R1 to R4
Next, write Tg
n
m + 1 in terms of the Mellin Transform for
Finite Abelian Groups and the Mellin Inversion Formula for
Finite Abelian Group to get Tg
n
m + 1 = Tg 1 + pν n1
m =
1
ϕ(pβ−ν
) χ(mod pβ−ν ) ˜gβ−ν(χ) χ n1
m .
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
123. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 2: Taking F4 from R1 to R4
Next, write Tg
n
m + 1 in terms of the Mellin Transform for
Finite Abelian Groups and the Mellin Inversion Formula for
Finite Abelian Group to get Tg
n
m + 1 = Tg 1 + pν n1
m =
1
ϕ(pβ−ν
) χ(mod pβ−ν ) ˜gβ−ν(χ) χ n1
m .
Substitution and simplification give
0≤ν<β
pν
2πi (σ)
F(s, v)p−νs
χ(mod pβ−ν )
˜gβ−ν(χ)
n1
(n1,pβ)=1
χ(n1)
ns
1
(m,pβ)=1
χ(m)
mu+v−s
ds.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
124. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 3: Taking F4 from R1 to R4
Applying the definition of the beta function and L-functions,
and taking σ = 2 will yield
0≤ν<β
pν
2πi
1
Γ(v) (2)
Γ(s)Γ(v − s)p−νs
χ(mod pβ−ν )
˜gβ−ν(χ)L(s, χ)
L(u + v − s, χ) ds.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
125. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Moving the Integral and Providing a Continuation
from R4 to D = {u , v ∈ C | |u |, |v | < B}
Next, we will move the integral from Re(s) = 2 to
Re(s) = 3B where B is a sufficiently large real number. This
will give a meromorphic continuation of the previous equation
for F4 to D.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
126. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 2: Moving the Integral and Providing a Continuation
from R4 to D
By using the residue theorem (F4,1 comes from the residue at s = u + v − 1,
F4,2 comes from the residues when there are nonpositive arguments for the
Gamma function, and F4,3 comes from the vertical strip Re(s) = 3B) we get
that F4(u, v; g) = − 0≤ν<β
pν
Γ(v)
F4,1(u, v; g) + F4,2(u, v; g) − F4,3(u, v; g)
where
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
127. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 2: Moving the Integral and Providing a Continuation
from R4 to D
By using the residue theorem (F4,1 comes from the residue at s = u + v − 1,
F4,2 comes from the residues when there are nonpositive arguments for the
Gamma function, and F4,3 comes from the vertical strip Re(s) = 3B) we get
that F4(u, v; g) = − 0≤ν<β
pν
Γ(v)
F4,1(u, v; g) + F4,2(u, v; g) − F4,3(u, v; g)
where
F4,1(u, v; g) :=
ϕ(pβ−ν
)
pβ−ν p−ν(u+v−1)Γ(u+v −1)Γ(1−u)L(u+v −1, χ0)˜gβ−ν (χ0),
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
128. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 2: Moving the Integral and Providing a Continuation
from R4 to D
By using the residue theorem (F4,1 comes from the residue at s = u + v − 1,
F4,2 comes from the residues when there are nonpositive arguments for the
Gamma function, and F4,3 comes from the vertical strip Re(s) = 3B) we get
that F4(u, v; g) = − 0≤ν<β
pν
Γ(v)
F4,1(u, v; g) + F4,2(u, v; g) − F4,3(u, v; g)
where
F4,1(u, v; g) :=
ϕ(pβ−ν
)
pβ−ν p−ν(u+v−1)Γ(u+v −1)Γ(1−u)L(u+v −1, χ0)˜gβ−ν (χ0),
F4,2(u, v; g) := j∈Z
Re(v−3B)≤j≤0
Γ(v − j)
(−1)−j
Γ(1−j)
p−ν(v−j)
χ(mod pβ−ν ) L(v −
j, χ)L(u + j, χ)˜gβ−ν (χ), and
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
129. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 2: Moving the Integral and Providing a Continuation
from R4 to D
By using the residue theorem (F4,1 comes from the residue at s = u + v − 1,
F4,2 comes from the residues when there are nonpositive arguments for the
Gamma function, and F4,3 comes from the vertical strip Re(s) = 3B) we get
that F4(u, v; g) = − 0≤ν<β
pν
Γ(v)
F4,1(u, v; g) + F4,2(u, v; g) − F4,3(u, v; g)
where
F4,1(u, v; g) :=
ϕ(pβ−ν
)
pβ−ν p−ν(u+v−1)Γ(u+v −1)Γ(1−u)L(u+v −1, χ0)˜gβ−ν (χ0),
F4,2(u, v; g) := j∈Z
Re(v−3B)≤j≤0
Γ(v − j)
(−1)−j
Γ(1−j)
p−ν(v−j)
χ(mod pβ−ν ) L(v −
j, χ)L(u + j, χ)˜gβ−ν (χ), and
F4,3(u, v; g) := 1
2πi (3B) Γ(s)Γ(v − s)p−νs
χ(mod pβ−ν ) L(s, χ)L(u + v − s, χ)
˜gβ−ν (χ) ds.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
130. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Break up F4,3 into Two Summands
Let u, v ∈ D. Then, F4(u, v; g) =
− 0≤ν<β
pν
Γ(v)
(F4,1(u, v; g)+F4,2(u, v; g)−F4,3,1(u, v; g)−F4,3,2(u, v; g)),
where
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
131. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Break up F4,3 into Two Summands
Let u, v ∈ D. Then, F4(u, v; g) =
− 0≤ν<β
pν
Γ(v)
(F4,1(u, v; g)+F4,2(u, v; g)−F4,3,1(u, v; g)−F4,3,2(u, v; g)),
where
F4,3,1(u, v; g) := 1
2πi (3B)
Γ(s)Γ(v − s)p−νs ∗
χ(mod pβ−ν )
τ(χ)
iaχ π1/2
pβ−ν
π
s−u−v
˜gβ−ν (χ)Γ
s−u−v+aχ+1
2
Γ
u+v−s+aχ
2
−1
∞
b=1
χ(b)
bs σu+v−1(b) ds
and
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
132. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Break up F4,3 into Two Summands
Let u, v ∈ D. Then, F4(u, v; g) =
− 0≤ν<β
pν
Γ(v)
(F4,1(u, v; g)+F4,2(u, v; g)−F4,3,1(u, v; g)−F4,3,2(u, v; g)),
where
F4,3,1(u, v; g) := 1
2πi (3B)
Γ(s)Γ(v − s)p−νs ∗
χ(mod pβ−ν )
τ(χ)
iaχ π1/2
pβ−ν
π
s−u−v
˜gβ−ν (χ)Γ
s−u−v+aχ+1
2
Γ
u+v−s+aχ
2
−1
∞
b=1
χ(b)
bs σu+v−1(b) ds
and
F4,3,2(u, v; g) := 1
2πi (3B)
Γ(s)Γ(v − s)p−νs β−ν
γ=1 χpγ ≡χ
τ(χpγ )
i
aχpγ
π1/2
pβ−ν
π
s−u−v
˜gβ−ν (χpγ )Γ
s−u−v+aχpγ +1
2
Γ
u+v−s+aχpγ
2
−1
∞
c=1
χpγ (c)
cs
σu+v−1(c) ds.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
133. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Break up F4,3 into Two Summands
Let u, v ∈ D. Then, F4(u, v; g) =
− 0≤ν<β
pν
Γ(v)
(F4,1(u, v; g)+F4,2(u, v; g)−F4,3,1(u, v; g)−F4,3,2(u, v; g)),
where
F4,3,1(u, v; g) := 1
2πi (3B)
Γ(s)Γ(v − s)p−νs ∗
χ(mod pβ−ν )
τ(χ)
iaχ π1/2
pβ−ν
π
s−u−v
˜gβ−ν (χ)Γ
s−u−v+aχ+1
2
Γ
u+v−s+aχ
2
−1
∞
b=1
χ(b)
bs σu+v−1(b) ds
and
F4,3,2(u, v; g) := 1
2πi (3B)
Γ(s)Γ(v − s)p−νs β−ν
γ=1 χpγ ≡χ
τ(χpγ )
i
aχpγ
π1/2
pβ−ν
π
s−u−v
˜gβ−ν (χpγ )Γ
s−u−v+aχpγ +1
2
Γ
u+v−s+aχpγ
2
−1
∞
c=1
χpγ (c)
cs
σu+v−1(c) ds.
Note that F4,3,1 uses the case that χ is primitive and F4,3,2 uses the case
that χ is not primitive. Then, use the functional equation to get the
result.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
134. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Symmetry of D
Since D = {u , v ∈ C | |u |, |v | < B} is symmetric we can take u = v = α for
0 < α < 1 to obtain
F4(α, α; g) = −
0≤ν<β
pν
Γ(α)
(F4,1(α, α; g) + F4,2(α, α; g)−
F4,3,1(α, α; g) − F4,3,2(α, α; g)).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
135. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Take The Integrals from Re(s) = 3B to Re(s) = 2
Let 0 < α < 1. Then,
F4(α, α; g) = −
0≤ν<β
pν
Γ(α)
(F4,1(α, α; g) + F4,2(α, α; g)−
F4,3,1,1(α, α; g) − F4,3,1,2(α, α; g) − F4,3,2,1(α, α; g) − F4,3,2,2(α, α; g)),
where
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
136. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Take The Integrals from Re(s) = 3B to Re(s) = 2
Let 0 < α < 1. Then,
F4(α, α; g) = −
0≤ν<β
pν
Γ(α)
(F4,1(α, α; g) + F4,2(α, α; g)−
F4,3,1,1(α, α; g) − F4,3,1,2(α, α; g) − F4,3,2,1(α, α; g) − F4,3,2,2(α, α; g)),
where
F4,3,1,1(α, α; g) consists of an integral and a sum over primitive characters
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
137. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Take The Integrals from Re(s) = 3B to Re(s) = 2
Let 0 < α < 1. Then,
F4(α, α; g) = −
0≤ν<β
pν
Γ(α)
(F4,1(α, α; g) + F4,2(α, α; g)−
F4,3,1,1(α, α; g) − F4,3,1,2(α, α; g) − F4,3,2,1(α, α; g) − F4,3,2,2(α, α; g)),
where
F4,3,1,1(α, α; g) consists of an integral and a sum over primitive characters
F4,3,1,2(α, α; g) consists of the residues and a sum over primitive
characters
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
138. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Take The Integrals from Re(s) = 3B to Re(s) = 2
Let 0 < α < 1. Then,
F4(α, α; g) = −
0≤ν<β
pν
Γ(α)
(F4,1(α, α; g) + F4,2(α, α; g)−
F4,3,1,1(α, α; g) − F4,3,1,2(α, α; g) − F4,3,2,1(α, α; g) − F4,3,2,2(α, α; g)),
where
F4,3,1,1(α, α; g) consists of an integral and a sum over primitive characters
F4,3,1,2(α, α; g) consists of the residues and a sum over primitive
characters
F4,3,2,1(α, α; g) consists of an integral and a sum over non-primitive
characters
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
139. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Take The Integrals from Re(s) = 3B to Re(s) = 2
Let 0 < α < 1. Then,
F4(α, α; g) = −
0≤ν<β
pν
Γ(α)
(F4,1(α, α; g) + F4,2(α, α; g)−
F4,3,1,1(α, α; g) − F4,3,1,2(α, α; g) − F4,3,2,1(α, α; g) − F4,3,2,2(α, α; g)),
where
F4,3,1,1(α, α; g) consists of an integral and a sum over primitive characters
F4,3,1,2(α, α; g) consists of the residues and a sum over primitive
characters
F4,3,2,1(α, α; g) consists of an integral and a sum over non-primitive
characters
F4,3,2,2(α, α; g) consists of the residues and a sum over non-primitive
characters
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
140. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
The Functional Equation and F2 Plus Simplification to F4
Let 0 < α < 1. Then,
F4(α, α; g) = −
0≤ν<β
pν
Γ(α)
(F4,1(α, α; g) + F4,2,1,1(α, α; g)+
F4,2,2,1(α, α; g) − F4,3,1,1(α, α; g) − F4,3,2,1(α, α; g)),
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
141. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
The Functional Equation and F2 Plus Simplification to F4
Let 0 < α < 1. Then,
F4(α, α; g) = −
0≤ν<β
pν
Γ(α)
(F4,1(α, α; g) + F4,2,1,1(α, α; g)+
F4,2,2,1(α, α; g) − F4,3,1,1(α, α; g) − F4,3,2,1(α, α; g)),
where F4,2,1,1 = F4,2,1 − F4,3,1,2 and F4,2,1 contains 2 more residues than
F4,3,1,2 and a sum over primitive characters
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
142. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
The Functional Equation and F2 Plus Simplification to F4
Let 0 < α < 1. Then,
F4(α, α; g) = −
0≤ν<β
pν
Γ(α)
(F4,1(α, α; g) + F4,2,1,1(α, α; g)+
F4,2,2,1(α, α; g) − F4,3,1,1(α, α; g) − F4,3,2,1(α, α; g)),
where F4,2,1,1 = F4,2,1 − F4,3,1,2 and F4,2,1 contains 2 more residues than
F4,3,1,2 and a sum over primitive characters
where F4,2,2,1 = F4,2,2 − F4,3,2,2 and F4,2,2 contains 2 more residues than
F4,3,2,2 and a sum over non-primitive characters
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
143. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Determining F4 + F5 as α → 1
2
Since F5(α, α; g) = F4(α, α; g), α ∈ R, and g is real-valued, it follows that
F4(α, α; g) + F5(α, α; g) = 2 Re F4(α, α; g).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
144. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
limα→1
2
2 Re F4,1
Note that
lim
α→ 1
2
2 Re F4,1(α, α; g) = π1/2 1 − p
p
Re (˜gβ−ν (χ0)) log p.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
145. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
limα→1
2
2 Re F4,1
Note that
lim
α→ 1
2
2 Re F4,1(α, α; g) = π1/2 1 − p
p
Re (˜gβ−ν (χ0)) log p.
We used Γ(2α − 1) = 1
2α−1
+ γ + O(|2α − 1|), l’Hˆopital’s rule, and
L(2α − 1, χ0) = ζ(2α − 1)
p|pβ−ν
p prime
1 − p1−2α
.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
146. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
limα→1
2
2 Re F4,2,1,1(α, α; g)
Note that
lim
α→ 1
2
2 Re F4,2,1,1(α, α; g) = 2
∞
d0=1
d(d0)d
−1/2
0
j=−1,−2
dj
0Γ
1
2
− j
(−1)−j
Γ(1 − j)
p−ν(1
2
−j) ∗
χ(mod pβ−ν )
π−1/2
Re i−aχ
τ(χ)˜gβ−ν (χ)χ(d0)
Γ
1
2
− j + aχ
2
Γ
1
2
+ j + aχ
2
−1
.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
147. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
limα→1
2
2 Re F4,2,1,1(α, α; g)
Note that
lim
α→ 1
2
2 Re F4,2,1,1(α, α; g) = 2
∞
d0=1
d(d0)d
−1/2
0
j=−1,−2
dj
0Γ
1
2
− j
(−1)−j
Γ(1 − j)
p−ν(1
2
−j) ∗
χ(mod pβ−ν )
π−1/2
Re i−aχ
τ(χ)˜gβ−ν (χ)χ(d0)
Γ
1
2
− j + aχ
2
Γ
1
2
+ j + aχ
2
−1
.
We checked for convergence and then we were able to take the limit
inside of the sum.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
148. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
limα→1
2
2 Re F4,2,2,1(α, α; g)
Note that
lim
α→ 1
2
2 Re F4,2,2,1(α, α; g) =
∞
f =1
2d(f )f −1/2
j=−2,−1
f j
Γ
1
2
− j
(−1)−j
Γ(1 − j)
p−ν(1
2
−j) pβ−ν
π
− 1
2
−j β−ν
γ=1 χpγ ≡χ
π−1/2
Γ
aχpγ − j + 1
2
2
Γ
aχpγ + 1
2
+ j
2
−1
Re i
−aχpγ
τ(χpγ )˜gβ−ν (χpγ )χpγ (f ) .
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
149. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
limα→1
2
2 Re F4,2,2,1(α, α; g)
Note that
lim
α→ 1
2
2 Re F4,2,2,1(α, α; g) =
∞
f =1
2d(f )f −1/2
j=−2,−1
f j
Γ
1
2
− j
(−1)−j
Γ(1 − j)
p−ν(1
2
−j) pβ−ν
π
− 1
2
−j β−ν
γ=1 χpγ ≡χ
π−1/2
Γ
aχpγ − j + 1
2
2
Γ
aχpγ + 1
2
+ j
2
−1
Re i
−aχpγ
τ(χpγ )˜gβ−ν (χpγ )χpγ (f ) .
We checked for convergence and then we were able to take the limit
inside of the sum.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
150. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Introduce F4,3,1,1,even and F4,3,1,1,odd
Let
F4,3,1,1,even(α, α; g) =
1
2πi
∞
b=1
σ2α−1(b)
(2)
b−s pβ−ν
π
s−2α
Γ(s)
Γ(α − s)p−νs
Γ
s − 2α + 1
2
Γ
2α − s
2
−1
ds
∗
χ(mod pβ−ν
)
χ even
τ(χ)
π1/2
˜gβ−ν (χ)χ(b)
and
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
151. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 2: Introduce F4,3,1,1,even(α, α; g) and
F4,3,1,1,odd(α, α; g)
F4,3,1,1,odd(α, α; g) =
1
2πi
∞
b=1
σ2α−1(b)
(2)
b−s pβ−ν
π
s−2α
Γ(s)
Γ(α − s)p−νs
Γ
s − 2α + 2
2
Γ
2α − s + 1
2
−1
ds
∗
χ(mod pβ−ν
)
χ odd
τ(χ)
iπ1/2
˜gβ−ν (χ)χ(b).
Then, we have that F4,3,1,1(α, α; g) = F4,3,1,1,even(α, α; g) + F4,3,1,1,odd(α, α; g).
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
152. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
limα→1
2
2 Re F4,3,1,1,even(α, α; g)
Note that
lim
α→ 1
2
2 Re F4,3,1,1,even(α, α; g) =
1
π1/2pβ−ν
∞
b=1
d(b) Im W1(b)
∗
χ(mod pβ
)
χ even
τ(χ)˜gβ−ν (χ)χ(b)
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
153. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
limα→1
2
2 Re F4,3,1,1,even(α, α; g)
Note that
lim
α→ 1
2
2 Re F4,3,1,1,even(α, α; g) =
1
π1/2pβ−ν
∞
b=1
d(b) Im W1(b)
∗
χ(mod pβ
)
χ even
τ(χ)˜gβ−ν (χ)χ(b)
where
W1(b) = π−1/2
(2)
2bπ
pβ−ν
−s
Γ2
(s)Γ
1
2
− s e−isπ/2
ds +
(2)
2bπ
pβ−ν
−s
Γ2
(s)Γ
1
2
− s eisπ/2
ds .
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
154. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
limα→1
2
2 Re F4,3,1,1,odd(α, α; g)
Note that
lim
α→ 1
2
2 Re F4,3,1,1,odd(α, α; g) =
1
π1/2pβ−ν
∞
b=1
d(b) Re W2(b)
∗
χ(mod pβ
)
χ odd
τ(χ)˜g(χ)χ(b) .
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
155. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
limα→1
2
2 Re F4,3,1,1,odd(α, α; g)
Note that
lim
α→ 1
2
2 Re F4,3,1,1,odd(α, α; g) =
1
π1/2pβ−ν
∞
b=1
d(b) Re W2(b)
∗
χ(mod pβ
)
χ odd
τ(χ)˜g(χ)χ(b) .
where
W2(b) = π1/2
i
(2)
2bπ
pβ−ν
−s
Γ2
(s)Γ
1
2
− s e−isπ/2
ds −
(2)
2bπ
pβ−ν
−s
Γ2
(s)Γ
1
2
− s eisπ/2
ds .
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
156. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
limα→1
2
2 Re F4,3,2,1(α, α; g)
Note that
lim
α→ 1
2
2 Re F4,3,2,1(α, α; g) =
1
π1/2pβ−ν
∞
c=1
d(c) Im W1(c)
β−ν
γ=1
χpγ ≡χ
χpγ even
τ(χpγ )˜gβ−ν (χpγ )χpγ (c) + Re W2(c)
β−ν
γ=1 χpγ ≡χ
χpγ odd
τ(χpγ )
˜gβ−ν (χpγ )χpγ (c) .
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
157. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Bring The Last Three Summands of the Main
Theorem Together
Note that
A2(g) = −
0≤ν<β
pν
π1/2
lim
α→ 1
2
2 Re F4,1(α, α; g),
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
158. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Bring The Last Three Summands of the Main
Theorem Together
Note that
A2(g) = −
0≤ν<β
pν
π1/2
lim
α→ 1
2
2 Re F4,1(α, α; g),
A3(g) = −
0≤ν<β
pν
π1/2
lim
α→ 1
2
2 Re F4,2,1,1(α, α; g)+ lim
α→ 1
2
2 Re F4,2,2,1(α, α; g) ,
and
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
159. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Part 1: Bring The Last Three Summands of the Main
Theorem Together
Note that
A2(g) = −
0≤ν<β
pν
π1/2
lim
α→ 1
2
2 Re F4,1(α, α; g),
A3(g) = −
0≤ν<β
pν
π1/2
lim
α→ 1
2
2 Re F4,2,1,1(α, α; g)+ lim
α→ 1
2
2 Re F4,2,2,1(α, α; g) ,
and
A4(g) = −
0≤ν<β
pν
π1/2
− lim
α→ 1
2
2 Re F4,3,1,1(α, α; g)− lim
α→ 1
2
2 Re F4,3,2,1(α, α; g) .
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
160. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Closing Thoughts
We have a formula which consists of a main term from the diagonal and
near-diagonal terms as well as three lesser terms from the off-diagonal
terms which can be expressed in terms of residues and integrals.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
161. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Closing Thoughts
We have a formula which consists of a main term from the diagonal and
near-diagonal terms as well as three lesser terms from the off-diagonal
terms which can be expressed in terms of residues and integrals.
There is a strong correspondence with Motohashi’s formula in which the
divisor function and transforms appear in similar locations.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
162. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Closing Thoughts
We have a formula which consists of a main term from the diagonal and
near-diagonal terms as well as three lesser terms from the off-diagonal
terms which can be expressed in terms of residues and integrals.
There is a strong correspondence with Motohashi’s formula in which the
divisor function and transforms appear in similar locations.
Now, I am currently investigating the asymptotics for particular test
functions and seeing how these correspond with the asymptotics in
Motohashi’s formula.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
163. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Closing Thoughts
We have a formula which consists of a main term from the diagonal and
near-diagonal terms as well as three lesser terms from the off-diagonal
terms which can be expressed in terms of residues and integrals.
There is a strong correspondence with Motohashi’s formula in which the
divisor function and transforms appear in similar locations.
Now, I am currently investigating the asymptotics for particular test
functions and seeing how these correspond with the asymptotics in
Motohashi’s formula.
In the future, it might be of interest to determine what would happen for
other power moments.
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ
164. Outline
Introduction
Main Theorem
Five Summands
Diagonal Sum
Near-Diagonal Sums
Off-Diagonal Sums
How We Get A2(g), A3(g), and A4(g)
Thank you
Thank you very much for listening!
Frank Romascavage, III Mean Square of a Dirichlet L-function of Modulus pβ