# 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)

containing two Dirichlet series.

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 TwoSeriesArisingFromTheAlternatingZetaFunction 2013-06-06 19:13:30 2013-06-06 19:13:30 pahio (2872) pahio (2872) 10 pahio (2872) Conjecture msc 30D99 msc 30B50 msc 11M41 trigonometric series conjecture equivalent to the Riemann hypothesis EulerRelation