Why the Riemann Xi function has only real zeros - using 2 integration approaches to Riemann's original equation to create a series of incomplete gamma functions and a power series and investigating the behaviour of the curves and the zeros when varying components. Conclusion - only real zeros (Riemann Hypothesis is true).
1. A proof of the Riemann Hypothesis - Highlights
A.A. Logan
The Riemann Xi function has only real zeros - using Incomplete Gamma Function
Series and Power Series Approaches
September 2018
2. Introduction
Upper Incomplete Gamma Functions Series (continued
fractions - behaviour when varying components).
Power Series (Behaviour of real and imaginary zeros)
Conclusion - all real zeros
Full paper at http://vixra.org/abs/1802.0124
Including detailed arguments.
Contact andrewalogan@gmail.com
3. Upper Incomplete Gamma Functions Series
Riemann’s Original Equation:
ξ(t)=4
∞
1 (d/dx(x3/2ψ (x)))x−1/4cos(t
2logx)dx, where
ψ(x) = ∞
m=1 e−m2πx
Using: t/2=a+bi, cos(x) = 1
2(eix + e−ix ) and y=m2πx, so
that dy=m2πdx:
ξ(t)/4= ∞
m=1
1
2(m2π)−1
4 ((m2π)(b−ai)Γ(9
4 − b + ai, m2π) +
(m2π)(−b+ai)Γ(9
4 + b − ai, m2π)) +
∞
m=1 −3
4(m2π)−1
4 ((m2π)(b−ai)Γ(5
4 − b + ai, m2π) +
(m2π)(−b+ai)Γ(5
4 + b − ai, m2π))
Looking at only -1/2 < b < +1/2 (Riemann proved no zeros
outside this limit).
With fixed b and increasing a, individual functions no zeros,
decreasing and tending to zero in the limit. Whole function
tends to zero in the limit.
As b increases, magnitude of Γ(5/4 + b − ai, m2π) increases.
See figure 1 below.
6. Continued Fractions and other components continued
Continued Fractions (CF) magnitude less than 1, decreasing
for increasing x, no zeros (and real and imaginary parts no
zeros).
Comparing CF (5/4+b-ai) and CF(5/4-b+ai), they both have
positive real parts but imaginary parts of different signs.
When b=0, the imaginary parts sum to 0.
When b=0, the magnitude of the CF increases as b increases.
NB the decrease in magnitude for (-b) is greater than the
increase in magnitude for (+b). See figures for m=1 - figure
2, m=2 - figure 3 and m=3 - figure 4.
The (m2π)e−m2π components. As m increases, these
components are real and reduce in magnitude very rapidly and
consistently.
The (−1
8 − b
2 + a
2i) and (−1
8 + b
2 − a
2i) + 1
2) components as
seen in figure 5 below (increasing in magnitude almost
linearly):
7. Continued Fraction m=1 - Illustration
Figure 2: Plots of Continued Fraction Components m=1.
8. Continued Fraction m=2 - Illustration
Figure 3: Plots of Continued Fraction Components m=2.
9. Continued Fraction m=3 - Illustration
Figure 4: Plots of Continued Fraction Components m=3.
10. (−1
8 − b
2 + a
2i) and (−1
8 + b
2 − a
2i) - Illustration
Figure 5: Plots of (−1
8 − b
2 + a
2 i) and (−1
8 + b
2 − a
2 i)
11. Combining the Components
Combining the above components, if the CF real and
imaginary parts are large enough, then the complete expression
can have a negative real part for large enough imaginary part
of s (positive for small imaginary part of s, going negative and
then tending to zero from below). If not large enough, then
always positive and tending to zero from above.
m=1 - ’large enough’ - so goes negative and stays below the
line (1 zero).
m=2,3,4... not large enough - always positive (and for m
increasing, magnitude decreasing rapidly).
Can only generate zeros by adding all the terms. If for any m
when added, no additional zeros generated, then no further
zeros generated by higher m components.
In addition, zeros generated by the shape of the m=1 curve.
12. Combining the Components continued
Considering the relative magnitudes of the CF terms and the
(-1/8-/+b/2+/-ai/2) terms, when b=0, then when they are
multiplied and added the sum will be greater than when b=0
(for both real and imaginary components).
Key point - this means that for the combined curve, when
b=0, there will be no additional zeros on the real curve
compared with when b=0.
This is because, since the magnitude of the curve increases as
b increases, positive points on the curve will stay positive (and
become greater) and negative points will stay negative (and
become greater).
To reiterate, no additional zeros on the real curve when b=0 -
ie no additional zeros with an imaginary component!
This is what we observe in the actual curves - see figures 6, 7
and 8 below:
16. Power Series
Riemann’s Original Equation:
ξ(t)=4
∞
1 (d/dx(x3/2ψ (x)))x−1/4cos(t
2logx)dx, where
ψ(x) = ∞
m=1 e−m2πx
To avoid ξ-Ξ confusion, the equation from Edwards is used:
ξ(s)=4
∞
1 (d/dx(x3/2ψ (x)))x−1/4cosh(1
2(s − 1
2)logx)dx
Now, if t=(x+yi), then (s-1
2) = it = (xi-y), and:
ξ(s)= ∞
n=0 a2n(xi − y)2n (all a2n positive)
This has been proven by Hadamard to converge (rapidly).
For x=0, no zeros. For y=0, potentially many zeros.
17. Riemann Xi Components
Riemann’s Xi function Definition: ξ(s) = Π(s
2)(s − 1)π
−s
2 ζ(s)
The only zeros are generated by the ζ(s) part.
Due to Euler’s
Π(s) = lim
N→∞
N!
(s + 1)(s + 2)...(s + N)
(N + 1)s
We know that the whole function decreases in magnitude for
increasing imaginary component and tends to zero in the limit.
We also know that there are real zeros all the way to the limit.
18. Polar Coordinates and paths of real part zeros
(xi − y)2n can be rewritten as r2n(cosθ + isinθ)2n and
expanded as r2ncos2nθ + ir2nsin2nθ, so:
ξ(s)= ∞
n=0 a2nr2n(cosθ + isinθ)2n
Using θ = (π
2 + )
cos2θ = cos(2(π
2 + )) = cosπcos2 - sinπsin2 = - cos2 and
cos(2(π
2 − )) = cosπcos2 + sinπsin2 = - cos2
Similar expressions can be generated for cos2nθ for all values
of n with similar results (except alternating signs).
a0 + a2r2cos2θ + a4r4cos4θ + a6r6cos6θ + a8r8cos8θ...=0 is
reflected across θ = π
2 for varying r
Real part zeros: family of curves, no intersections between
members of family (single valued).
19. Polar Coordinates and paths of imaginary part zeros
Using θ = (π
2 + ):
sin2θ = sin(2(π
2 + )) = sinπcos2 + cosπsin2 = - sin2 and
sin(2(π
2 − )) = sinπcos2 - cosπsin2 = + sin2
Similar expressions can be generated for sin2nθ for all values
of n with similar results (except alternating signs).
a2r2sin2θ + a4r4sin4θ + a6r6sin6θ + a8r8sin8θ...=0 is
reflected across θ = π
2 for varying r.
Imaginary part zeros: family of curves, no intersections
between members of family (single valued).
When θ = π
2 , then we know that the function is identically
zero.
See figure 9 below for illustration.
20. Real and Imaginary part zero paths
Figure 9: Paths of real part and imaginary part zeros.
21. (Real+Imag) and (Real-Imag) part zeros
The complete function will be zero when both real and
imaginary expressions are equal to each other and both zero.
Thus we are looking for common zeros of these two
expressions:
a0 + a2r2cos2θ + a4r4cos4θ + a6r6cos6θ + a8r8cos8θ...=0 (1)
a2r2sin2θ + a4r4sin4θ + a6r6sin6θ + a8r8sin8θ...=0 (2)
Combining ((1)+(2)) and ((1)-(2)). If and only if they are
simultaneously zero then the complete function is zero.
The expressions (1)+(2) and (1)-(2) are reflected through
θ = π
2 and if they cross θ = π
2 then there will be a coincident
pair of real zeros on θ = π
2 .
Again single valued, no intersections with other curves in the
family.
This means that there will be no additional complete function
zeros generated - each pair of imaginary part zeros will only
coincide with one pair of real part zeros.
See figure 10 for illustration.
23. Conclusions
Known initially - no zeros outside the critical strip
Shown with incomplete gamma function/CF approach - no
additional zeros when b ¿ 0 than when b = 0.
Shown with Xi function components - real zeros all the way to
the limit.
Shown with polar coordinates and zero paths - no additional
whole function zeros generated for any variable values.
Combining the above - the Riemann Xi function has only real
zeros - the Riemann Hypothesis is true.
