two series arising from the alternating zeta function

The terms of the series defining the alternating zeta function

η(s):=n=1(-1)n-1ns  (Res>0),

a.k.a. the Dirichlet eta functionMathworldPlanetmath, may be split into their real and imaginary partsDlmfMathworldPlanetmath:


Here,  s=a+ib  with real a and b.  It follows the equation

η(s)=-n=1(-1)nnacos(blnn)+in=1(-1)nnasin(blnn) (1)

containing two Dirichlet series.

The alternating zeta function and the Riemann zeta functionDlmfDlmfMathworldPlanetmath are connected by the relation


(see the parent entry (  The following conjecture concerning the above real part series and imaginary part series of (1) has been proved by Sondow [1] to be equivalent with the Riemann hypothesis.

Conjecture.  If the equations

n=1(-1)nnacos(blnn)= 0andn=1(-1)nnasin(blnn)= 0

are true for some pair of real numbers a and b, then

a= 1/2  or  a= 1.


  • 1 Jonathan Sondow: A simple counterexample to Havil’s “reformulation” of the Riemann hypothesis.  – Elemente der Mathematik 67 (2012) 61–67.  Also available
Title two series arising from the alternating zeta function
Canonical name TwoSeriesArisingFromTheAlternatingZetaFunction
Date of creation 2013-06-06 19:13:30
Last modified on 2013-06-06 19:13:30
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Conjecture
Classification msc 30D99
Classification msc 30B50
Classification msc 11M41
Synonym trigonometric series conjecture equivalent to the Riemann hypothesis
Related topic EulerRelation