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arXiv · 2312.05138

M\"obius function and primes: an identity factory with applications

Olivier Ramaré·Sebastian Zuniga Alterman
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Abstract

We investigate the sums $\sum_{n\le X, (n,q)=1}\frac{\mu(n)}{n^s}\log^k\left(\frac{X}{n}\right)$, where $k\in\{0,1\}$, $s\in\mathbb{C}$, $\Re s>0$. Our goal is to obtain explicit asymptotic estimations for these quantities. To achieve this, we develop a broad framework of identities that we use to derive several applications. Building on similar principles, we also provide an appendix establishing the inequality $\sum_{n\le X}\Lambda(n)/n\le \log X$, valid for any $X\geq 1$.

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Olivier Ramaré, Sebastian Zuniga Alterman. 2023-12-08. M\"obius function and primes: an identity factory with applications. https://arxiv.org/abs/2312.05138

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