Distribution and Non-vanishing of special values of $L$-series attached to Erdős functions

Abstract

In a written correspondence with A. Livingston, Erdős conjectured that for any arithmetical function $f$, periodic with period $q$, taking values in $\{-1,1\}$ when $q\nmid n$ and $f(n)=0$ when $q\mid n$, the series ${\sum }_{n=1}^{\infty }f(n)/n$ does not vanish. This conjecture is still open in the case $q\equiv 1\mathrm{mod}4$ or when $2\varphi (q)+1\le q$. In this paper, we obtain the characteristic function of the limiting distribution of $L(k,f)$ for any positive integer $k$ and Erdős function $f$ with the same parity as $k$. Moreover, we show that the Erdős conjecture is true with "probability" one.

12 pages