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