formulae for zeta in the critical strip

Let us use the traditional notation s=σ+it for the complex variable, where σ and t are real numbers.

ζ(s) = 11-21-sn=1(-1)n+1n-s  σ>0 (1)
ζ(s) = 1s-1+1-s1x-[x]xs+1𝑑x  σ>0 (2)
ζ(s) = 1s-1+12-s1((x))xs+1𝑑xσ>-1 (3)

where [x] denotes the largest integer x, and ((x)) denotes x-[x]-12.

We will prove (2) and (3) with the help of this useful lemma:

Lemma: For integers u and v such that 0<u<v:


Proof: If we can prove the special case v=u+1, namely

(u+1)-s=-suu+1x-[x]xs+1𝑑x+(u+1)1-s-u1-s1-s (4)

then the lemma will follow by summing a finite sequencePlanetmathPlanetmath of cases of (4). The integral in (4) is

01tdt(u+t)s+1 = 01(u+t)-s𝑑t-01u(u+t)-s-1𝑑t
= (u+1)1-s-u1-s1-s+u[(u+1)-s-u-s]s

so the right side of (4) is


and the lemma is proved.

Now take u=1 and let v in the lemma, showing that (2) holds for σ>1. By the principle of analytic continuation, if the integral in (2) is analytic for σ>0, then (2) holds for σ>0. But x-[x] is boundedPlanetmathPlanetmathPlanetmathPlanetmath, so the integral converges uniformly on σϵ for any ϵ>0, and the claim (2) follows.

We have


Adding and subtracting this quantity from (2), we get (3) for σ>0. We need to show that


is analytic on σ>-1. Write


and integrate by parts:


The first two terms on the right are zero, and the integral convergesPlanetmathPlanetmath for σ>-1 because f is bounded.

Remarks: We will prove (1) in a later version of this entry.

Using formulaMathworldPlanetmathPlanetmath (3), one can verify Riemann’s functional equation in the strip -1<σ<2. By analytic continuation, it follows that the functional equation holds everywhere. One way to prove it in the strip is to decompose the sawtooth function ((x)) into a Fourier series, and do a termwise integration. But the proof gets rather technical, because that series does not converge uniformly.

Title formulae for zeta in the critical stripMathworldPlanetmath
Canonical name FormulaeForZetaInTheCriticalStrip
Date of creation 2013-03-22 13:28:14
Last modified on 2013-03-22 13:28:14
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 11
Author mathcam (2727)
Entry type Theorem
Classification msc 11M99
Related topic CriticalStrip
Related topic ValueOfTheRiemannZetaFunctionAtS0
Related topic AnalyticContinuationOfRiemannZeta