two series arising from the alternating zeta function

The terms of the series defining the alternating zeta function

\eta(s)\;:=\;\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{s}}\qquad(\mbox{Re}\,s>0),

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

\frac{1}{n^{s}}\;=\;\frac{e^{-ib\ln{n}}}{n^{a}}\;=\;\frac{\cos(b\ln{n})}{n^{a}% }-\frac{i\sin(b\ln{n})}{n^{a}}

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

\displaystyle\eta(s)\;=\;-\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{a}}\cos(b\ln{n% })+i\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{a}}\sin(b\ln{n})

(1)

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

\zeta(s)\;=\;\frac{\eta(s)}{1-2^{1-s}}

(see the parent entry (http://planetmath.org/AnalyticContinuationOfRiemannZetaToCriticalStrip)). 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

\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{a}}\cos(b\ln{n})\;=\;0\quad\mbox{and}% \quad\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{a}}\sin(b\ln{n})\;=\;0

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

a\;=\;1/2\qquad\mbox{or}\qquad a\;=\;1.

References

1 Jonathan Sondow: A simple counterexample to Havil’s “reformulation” of the Riemann hypothesis. – Elemente der Mathematik 67 (2012) 61–67. Also available http://arxiv.org/pdf/0706.2840v3.pdfhere.

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