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arXiv · 2504.16002

A logarithmic analogue of Alladi's formula

Biao Wang
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Abstract

Let $\mu(n)$ be the M\"{o}bius function. Let $P^-(n)$ denote the smallest prime factor of an integer $n$. In 1977, Alladi established the following formula related to the prime number theorem for arithmetic progressions \[ -\sum_{\substack{n\geq 2\\ P^-(n)\equiv \ell ({\rm mod}k)}}\frac{\mu(n)}{n}=\frac1{\varphi(k)} \] for positive integers $\ell, k\ge$ with $(\ell,k)=1$, where $\varphi$ is Euler's totient function. In this note, we will show a logarithmic analogue of Alladi's formula in an elementary proof.

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Biao Wang. 2025-04-22. A logarithmic analogue of Alladi's formula. https://doi.org/10.1007/s11139-025-01212-5

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