convergence of Riemann zeta series

The series

n=11ns (1)

converges absolutely for all s with real partMathworldPlanetmath greater than 1.

Proof. Let  s=σ+it  where  σ and t are real numbers and  σ>1.  Then


Since the series  n=11nσ converges, by the p-test (, for  σ>1, we conclude that the series (1) is absolutely convergent in the half-plane  σ>1.

Title convergence of Riemann zeta series
Canonical name ConvergenceOfRiemannZetaSeries
Date of creation 2015-08-22 13:15:14
Last modified on 2015-08-22 13:15:14
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Definition
Classification msc 11M06
Classification msc 30A99
Related topic ModulusOfComplexNumber
Related topic ComplexExponentialFunction