Analytic Continuation of the Riemann Zeta Function
Posted November 28. 2020.
819 words.
4 min read.
In the study of analytic number theory, the Riemannn zeta function ζ(s) is a frequently used tool to study number-theoretic objects. Originally, ζ(s) is defined as
ζ(s)=n=1∑∞ns1(1)
To determine its convegence, let's consider Riemann-Stieltjes integration:
It can be easily verified that this expression converges absolutely and uniformly when ℜ(s)>1, which allows us to make some manipulations with it. Let's have a look
An Identity due to Poisson's Summation Formula
Define
ψ(x)=n=1∑∞e−n2πx
Then by Poisson's summation formula we have
2ψ(x)+1=n∈Z∑e−n2πx=x1n∈Z∑e−n2π/x
which leads to
ψ(x)=x1ψ(x1)+2x1−21(2)
Integral Representation for ζ(s)
Let's perform a Mellin transform on this function so that
As we can observe that the right hand side does not change when we replace s with 1−s. Hence, by (3) we have
π−s/2Γ(2s)ζ(s)=π(s−1)/2Γ(21−s)ζ(1−s)
Now, in order to achieve the optimal simplicity, we multiply both side with Γ(1−2s):
Γ(2s)Γ(1−2s)ζ(s)=πs−1/2Γ(21−s)Γ(21−s+21)ζ(s)
By Euler's reflection formula, we have
Γ(2s)Γ(1−2s)=πcsc(2πs)
and by Legendre's Duplication formula, we deduce
Γ(21−s)Γ(21−s+21)=2sπ1/2Γ(1−s)
Plugging these results back gives us
πcsc(2πs)ζ(s)=2sπsΓ(1−s)ζ(1−s)
Now, if we were to perform more manipulations, we get
ζ(s)=2sπs−1sin(2πs)Γ(1−s)ζ(1−s)
which is known as the functional equation for ζ(s).
Conclusion
In this blog post, we begin with the Dirichlet series definition of ζ(s), 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
ζ(s)=2sπs−1sin(2πs)Γ(1−s)ζ(1−s),
we can observe that for ℜ(s)<0 the right hand side becomes zero whenever s=−2k=0. Hence, we call such s's as the trivial zeros of ζ(s). 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 ζ(s).
On going through these definition, we can now have a good basic grasp of the Riemann hypothesis:
Riemann hypothesis: All nontrivial zeros of ζ(s) lie on the line ℜ(s)=21.