On a conjecture of Goldmakher

Abstract

We construct a $1$-bounded completely multiplicative function $f$ whose logarithmically-averaged partial sums satisfy $$\underset{x\to \infty }{\mathrm{limsup}}\frac{\left|{\sum }_{n\le x}\frac{f(n)}{n}\right|}{1+\exp \left({\sum }_{p\le x}\frac{\text{Re}(f(p))}{p}\right)}=\infty .$$ This disproves a conjecture of Goldmakher from 2009.

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