On multiplicative functions with small partial sums

Stelios Sachpazis

doi:10.1093/imrn/rnad071

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arXiv · 2208.06230

On multiplicative functions with small partial sums

Stelios Sachpazis

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Abstract

In analytic number theory, several results make use of information regarding the prime values of a multiplicative function in order to extract information about its averages. Examples of such results include Wirsing's theorem and the Landau-Selberg-Delange method. In this paper, we are interested in the opposite direction. In particular, we prove that when $f$ is a suitable divisor-bounded multiplicative function with small partial sums, then $f(p)\approx-p^{iγ_1}-\ldots-p^{iγ_m}$ on average, where the $γ_j$'s are the imaginary parts of the zeros of the Dirichet series of $f$ on the line $\Re(s)=1$. This extends a result of Koukoulopoulos and Soundararajan and it builds upon ideas coming from previous work of Koukoulopoulos for the case where $|f|\leqslant 1$.

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BibTeX RIS

Stelios Sachpazis. 2023-06-11. On multiplicative functions with small partial sums. https://doi.org/10.1093/imrn%2Frnad071

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