analytic continuation of Riemann zeta to critical strip

The   1ns=e-slogn  (see the general power) of the series

n=11ns= 1+12s+13s+14s+, (1)

defining the Riemann zeta functionDlmfDlmfMathworldPlanetmath ζ(s) for  s>1,  are holomorphic in the whole s-plane and the series converges uniformly in any closed disc of the half-plane  s>1 ((let  s=σ+it  with σ,t  and σ>1;  then |1ns|=1nσ1n1+d  for a positive d for all  n=1, 2,;  the series  n=11n1+d converges since  1+d>1;  thus the series (1) converges uniformly in the closed half-plane  s1+d,  by the Weierstrass criterion (  Therefore we can infer (see theorems on complex function series ( that the sum ζ(s) of (1) is holomorphic in the domain  s>1.

We use also the fact that the series

n=1(-1)n-1ns= 1-12s+13s-14s+- (2)

defining the Dirichlet eta functionMathworldPlanetmath η(s), a.k.a. the alternating zeta function, is convergentMathworldPlanetmathPlanetmath for  s>0  and its sum is holomorphic in this half-plane.

If we multiply the series (1) by the difference 1-22s, every other of the series changes its sign and we get the series (2).  So we can write

ζ(s)=η(s)1-22s, (3)

which is valid when the denominator does not vanish and  s>1.  The zeros of the denominator are obtained from  2s=2,  i.e. from


This gives  slog2-log2=n2iπ (see the periodicity of exponential functionDlmfDlmfMathworldPlanetmathPlanetmath), i.e.

s= 1+n2πiln2(n). (4)

Thus the zeros of the denominator of (3) are on the line  s=1.

Now the functionMathworldPlanetmath on the right hand side of (3) is holomorphic in the set


and the values of this function coincide with the values of zeta functionMathworldPlanetmath in the half-plane  s>1.

This result means that, via the equation (3), the zeta function has been analytically continued ( to the domain D, as far as to the imaginary axis.

Remark.  In reality, all points (4) except  s=1  are removable singularitiesMathworldPlanetmath of ζ(s) given by (3), due to the fact that they are also zeros of η(s).  The fact is considered in the entry zeros of Dirichlet eta function.

Charles Hermite has shown that the zeta function may be analytically continued to the whole s-plane except for a simple poleMathworldPlanetmathPlanetmath at  s=1,  by using the equation

ζ(s)=1Γ(s)0xs-1ex-1𝑑x. (5)

See this article (

Title analytic continuation of Riemann zeta to critical stripMathworldPlanetmath
Canonical name AnalyticContinuationOfRiemannZetaToCriticalStrip
Date of creation 2015-08-22 13:33:33
Last modified on 2015-08-22 13:33:33
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 23
Author pahio (2872)
Entry type Example
Classification msc 30D30
Classification msc 30B40
Classification msc 11M41
Related topic RiemannZetaFunction
Related topic AnalyticContinuation
Related topic MeromorphicExtension
Related topic CriticalStrip
Related topic AnalyticContinuationOfRiemannZetaUsingIntegral
Related topic FormulaeForZetaInTheCriticalStrip
Related topic GammaFunction
Defines alternating zeta function