In ancient cultures the favorite question of sishya to his guru was "Who Am I?" and in the end he learnt everything about Reality. If you want a similar question in mathematics, ask "Is Riemann Hypothesis true?", and you will learn almost everything about mathematics. This presentation gives an elementary introduction to zeta functions using natural functions.

1. Riemann Hypothesis
and
Natural Functions
verS

2. Riemann Zeta Function
Riemann has shown that the function
has
1. just one singularity at s=1
2. “trivial” zeros at -2, -4, -6, …
3. an infinite number of nontrivial zeros in
the “critical strip” between the points
(0,y) and (1,y)

3. Natural Functions
where, variable t is real, n = 1, 2, 3, … ,
and is the usual square root of minus one.
In my opinion a concept as important as the
natural numbers is the natural functions:

4. Complex Natural Functions
where and n = 1, 2, 3, …

5. Alternate Form
It is easy to see that complex natural
functions can also be written as
where

6. Generalized Form
where

7. Electromagnetic Waves
Note that the universe is filled up with
electromagnetic waves represented by
linear combinations of functions of the
form:

8. Natural Riemann Function
We define natural Riemann function as:

9. Riemann Zeta Function
It is easy to see that Riemann zeta
function and natural Riemann function
are related by the equation:
This equation says that you will get the
natural zeta function if you rotate the
Riemann zeta function by ninety degrees
clockwise.

10. Time Function Transform
Here Bromwich contour goes from left to
right and

11. Frequency Function Transform
These transforms show the symmetry
between the time domain and frequency
domain analyses used in communication
technology.

12. Riemann Impulse Function
We define Riemann impulse function in
terms of the Dirac delta function as:

13. Impulse Function Transform
From the definition of Riemann impulse
function, it is easy to see that

14. Prime Functions
For the sequence of primes 2, 3, 5, …
we write
respectively, and for each positive integer
k we define a prime function as follows:
where the symbolic summation takes all
those products of primes taken k at a
time.

15. Euler Zeta Function
We define Euler zeta function in terms of
the prime functions as:

16. Euler Product Formula
The celebrated Euler product formula
states that

17. Riemann Functional Equation
Riemann first gets rid of the singularity at
s=1 and the trivial zeros of the zeta function,
by forming the function
Then defining
he writes the functional equation as

18. Riemann Hypothesis
The deceptively simple looking Riemann
hypothesis states as a guess that all the
zeros of the natural zeta function lies on a
horizontal line half unit under the x-axis of
the complex plane.

19. Epilogue
Our attempt here has been to state some
of the well-known results about Riemann
zeta function as simple as possible, so
that novices are enticed to take an in-
depth look at the Riemann hypothesis.