Riemann zeta function has no zeros on s=0,1

This article shows that the Riemann zeta functionMathworldPlanetmath ζ(s) has no zeros along the lines s=0 or s=1. That implies that all nontrivial zeros of ζ(s) lie strictly within the critical stripMathworldPlanetmath 0<s<1. As the article points out, this is known to be equivalent to one version of the prime number theoremMathworldPlanetmath.

It can in fact be shown that ζ(s)0 for any s=σ+it with 0<σ<1 if


for some constant c. By using the functional equation


we have also that ζ(σ+it)0 if


Bounding the zeros of ζ(s) away from s=0, 1 leads to a version of the prime number theorem with more precise error terms.

Theorem 1

ζ(1+it)0 for tR.

Proof. Notice that for θ

02(1+cosθ)2=2cos2θ+4cosθ+2=3+4cosθ+cos(2θ) (1)

If σ=s>1, then ζ(σ+it)=p prime(1-p-σ-it)-1, so that

logζ(σ+it)=-p primelog(1-p-σ-it)=p primem=11mp-mσ-imt

and thus

log|ζ(σ+it)|=p primem=11mpmσcos(mtlogp)

since the log of the absolute valueMathworldPlanetmathPlanetmathPlanetmathPlanetmath is the real partMathworldPlanetmath of the log.

Using equation (1), we then have

3logζ(σ)+ 4log|ζ(σ+it)|+log|ζ(σ+i2t)|
=p primem=11mpmσ(3+4cos(mtlogp)+cos(2mtlogp))0

so that

ζ(σ)3|ζ(σ+it)|4|ζ(σ+it2)|1 for all σ>1,t (2)

But if ζ has a zero at σ+it0, then


since the first factor gives a pole (http://planetmath.org/Pole) of order 3 at 1 and the second factor gives a zero of order at least 4 at 1+it0. This contradicts equation (2).

Corollary 1

ζ(it)0 for tR.

Proof.  Use the functional equation


and set s=it. The theorem implies that the RHS is nonzero, so the LHS is as well. Thus ζ(s)0.

Title Riemann zeta function has no zeros on s=0,1
Canonical name RiemannZetaFunctionHasNoZerosOnReS01
Date of creation 2013-03-22 17:54:37
Last modified on 2013-03-22 17:54:37
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 5
Author rm50 (10146)
Entry type Theorem
Classification msc 11M06