In the study of analytic number theory, the Riemannn zeta function is a frequently used tool to study number-theoretic objects. Originally, is defined as
To determine its convegence, let's consider Riemann-Stieltjes integration:
It can be easily verified that this expression converges absolutely and uniformly when , which allows us to make some manipulations with it. Let's have a look
An Identity due to Poisson's Summation Formula
Then by Poisson's summation formula we have
which leads to
Integral Representation for
Let's perform a Mellin transform on this function so that
Now, by (1) we obtain this identity:
As a result, we can study the properties of the Riemann zeta function by digging deeper into the integral on the left hand side.
Analytic Continuation of the Integral
First, let's split the integral into two parts
Then, applying (2) to side gives
Now, plugging this result to the original equation gives
As we can observe that the right hand side does not change when we replace with . Hence, by (3) we have
Now, in order to achieve the optimal simplicity, we multiply both side with :
By Euler's reflection formula, we have
and by Legendre's Duplication formula, we deduce
Plugging these results back gives us
Now, if we were to perform more manipulations, we get
which is known as the functional equation for .
In this blog post, we begin with the Dirichlet series definition of , and then we try to connect zeta function with an integral representation. Subsequently, we use Poisson's summation formula to obtain its analytic continuation. However, this analytic continuation has other impacts. If we look back to the equation
we can observe that for the right hand side becomes zero whenever . Hence, we call such 's as the trivial zeros of . However, there are also other occasions in which the right hand side is zero. Alternatively, we call that kind of zeros the nontrivial zeros of .
On going through these definition, we can now have a good basic grasp of the Riemann hypothesis:
Riemann hypothesis: All nontrivial zeros of lie on the line .