On the multiplicative Erd\H{o}s discrepancy problem

TL;DR

This paper investigates a longstanding conjecture by Erd ext{"o}s on multiplicative functions, proving unboundedness of their partial sums under certain average conditions, and extends these results to related functions.

Contribution

It establishes new conditions under which the partial sums of multiplicative functions are unbounded, advancing understanding of Erd ext{"o}s's discrepancy problem.

Findings

If c > 0, partial sums of f are unbounded.

If c < 0, partial sums of μf are unbounded.

Extensions of the main results are discussed.

Abstract

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