On the multiplicative Erdős discrepancy problem

Abstract

As early as the 1930s, Pál Erdős conjectured that: {\em for any multiplicative function $f:\mathbb{N}\to \{-1,1\}$, the partial sums ${\sum }_{n\le x}f(n)$ are unbounded.} Considering this conjecture, in this paper we consider multiplicative functions $f$ satisfying $$\sum _{p\le x}f(p)=c\cdot \frac{x}{\log x}(1+o(1)).$$ We prove that if $c>0$ then the partial sums of $f$ are unbounded, and if $c<0$ then the partial sums of $\hat{I}¼f$ are unbounded. Extensions of this result are also discussed.

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