Distribution and Non-vanishing of special values of $L$-series attached to Erd\H{o}s functions

Abstract

In a written correspondence with A. Livingston, Erd\H{o}s 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 $q \equiv 1 \bmod 4$ or when $2 \phi(q)+ 1 \leq q$. In this paper, we obtain the characteristic function of the limiting distribution of $L(k,f)$ for any positive integer $k$ and Erd\H{o}s function $f$ with the same parity as $k$. Moreover, we show that the Erd\H{o}s conjecture is true with "probability" one.