# two series arising from the alternating zeta function

The terms of the series defining the alternating zeta function

 $\eta(s)\;:=\;\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{s}}\qquad(\mbox{Re}\,s>0),$
 $\frac{1}{n^{s}}\;=\;\frac{e^{-ib\ln{n}}}{n^{a}}\;=\;\frac{\cos(b\ln{n})}{n^{a}% }-\frac{i\sin(b\ln{n})}{n^{a}}$

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.

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

(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  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