A logarithmic analogue of Alladi's formula

Abstract

Let $\mu (n)$ be the Möbius 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 \[ -\sum_{\substack{n\geq 2\\ P^-(n)\equiv \ell ({\rm mod}k)}}\frac{μ(n)}{n}=\frac1{φ(k)} \] for positive integers $\ell ,k\ge$ with $(\ell ,k)=1$, where $\phi$ is Euler's totient function. In this note, we will show a logarithmic analogue of Alladi's formula in an elementary proof.

arXiv comment15 pages

References in corpus (2)

• Analogues of Alladi's formula
• Analogues of Alladi's formula over global function fields