Riemann’s ten-page-long paper “Über die Anzahl der Primzahen unter einer gegebener Gröβe” has great influence on modern number theory. In the paper, he established two important properties of the Riemann-zeta function
(the summation is absolutely convergent for ).
Riemann showed that
- (i) The Riemann-zeta function has an analytic continuation to the whole complex plane as a meromorphic function, with a simple pole of residue at .
- (ii) The Riemann-zeta function has the functional equation.
Here is the Gamma function defined by
(Another important property of the Riemann-zeta function is the Euler product
which can be proved by the fundamental theorem of arithmetic and basic analysis.)
Riemann gave two proofs of the functional equation. One of the proof, which we will follow, used the Poission summation formula.
1. The Poisson summation formula
is uniformly and absolutely convergent to .
For arbitrary continuous function on , generally it is not a periodic function and thus we cannot apply the theory of Fourier series. We can, however, sum up all the translation of , i.e., and get a function of period one. We need some extra conditions on to ensure that satisfies the conditions in the above proposition.
We need to introduce a common notation. Given two functions and , we write , if there exists an absolute constant such that for all .
a periodic function with period one and has a continuous derivative, i.e., it satisfies the condition of the previous proposition.
Proposition 4 Suppose satisfies the conditions of Proposition 3. Then
Recall the Fourier transform of is defined by
Proof: The -the Fourier coefficient of is given by
The proposition then follows from Proposition 2.
Putting in the proposition, we obtain:
Theorem 5 (The Poisson Summation Formula) Suppose satisfies the conditions of Proposition 3, then
Applying the Poisson summation formula to , we have
Corollary 6 Suppose satisfies the conditions of Proposition 3, for real number ,
2. The functional equation of the Riemann-zeta function
From now on . It is a self-dual function (i.e., ). First consider the integral (we will justify the absolute convergence for few lines below.)
The summation and the integral are absolutely convergent for . The above is then
Applying Corollary 6 to the second integral, the above is
Thus the integrals of (8) are absolutely convergent for all . Therefore the right hand side of (8) can be extended meromorphically to , with simple poles of residue and of residue. Since is an entire function with zeros , theorem 1 (i) follows. (Except the calculation of the residue at , exercise.) Obviously the right hand side of (8) remains unchanged if we replace by . Theorem 1 (ii) then follows.