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arXiv · 2610.22081

Logarithmic exponential sums with multiplicative coefficients

Haozhe Gou
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Abstract

Let $f$ be a $1$-bounded multiplicative function, and suppose that $|α-a/q|\leq 1/(qx)$, where $(a,q)=1$ and $q\leq x$. We prove that $$ \left|\sum_{n\leq x}\frac{f(n)}{n}e( nα)\right| \leq \left(\frac{4}{\sqrt{6}}+o(1)\right)\frac{\log x}{\sqrt{q}}+O(\log q). $$ Here the constant $4/\sqrt{6}$ is optimal. This settles two conjectures of Granville and Lamzouri recently. If $f$ is completely multiplicative, the constant improves to $1$. The main new ingredient is a two-prime Ramaré decomposition identity.

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Haozhe Gou. 2026-10-15. Logarithmic exponential sums with multiplicative coefficients. https://arxiv.org/abs/2610.22081

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