On a conjecture of Erdős and certain Dirichlet series

Tapas Chatterjee and M. Ram Murty

Vol. 275 (2015), No. 1, 103–113

DOI: 10.2140/pjm.2015.275.103

Abstract

Let $f:\mathbb{Z}∕q\mathbb{Z}\to \mathbb{Z}$ be such that $f\left(a\right)=\pm 1$ for $1\le a<q$, and $f\left(q\right)=0$. Then Erdős conjectured that ${\sum }_{n\ge 1}f\left(n\right)∕n\ne 0$. For $q$ even, it is easy to show that the conjecture is true. The case $q\equiv 3\left(mod4\right)$ was solved by Murty and Saradha. In this paper, we show that this conjecture is true for 82% of the remaining integers $q\equiv 1\left(mod4\right)$.

Keywords

Erdős conjecture, Okada's criterion, nonvanishing of Dirichlet series

Mathematical Subject Classification 2010

Primary: 11M06, 11M41

Secondary: 11M20

Milestones

Received: 20 February 2014

Revised: 5 November 2014

Accepted: 5 November 2014

Published: 12 April 2015