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arXiv · 1003.5388

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

Michael Coons
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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.

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Michael Coons. 2010-03-28. On the multiplicative Erd\H{o}s discrepancy problem. https://arxiv.org/abs/1003.5388

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