Generalized relations between arithmetic functions
Jean-Christophe Pain
TL;DR
The paper addresses the problem of expressing divisor sums involving $ ilde ext{mu}^2$ and $ ext{phi}$ as finite Euler products, using a multiplicative framework to generalize the Dineva identity. It develops a general construction principle for identities of the form $F(n)= extstyle\sum_{digm|n} ilde ext{mu}^2(d)H(d)$ and derives the one-parameter family $F(n)= extstyle\sum_{digm|n} rac{ ilde ext{mu}^2(d)}{ ext{phi}(d) ext{d}^s}= extstyle\prod_{pigm|n}igl(1+ rac{1}{(p-1)p^s}igr)$. It connects these identities to partial zeta functions and the Selberg sieve, illustrating the structure with several examples and alternative Euler-product representations. The results provide a systematic method to generate similar divisor-sum identities and suggest avenues for generalizations to other arithmetic functions.
Abstract
The aim of this article is to present in a self-contained way identities arising in elementary number theory, among which the following one: $$ \sum_{d\mid n}\frac{μ^2(d)}{\varphi(d)\,d^s}=\prod_{p\mid n}\left(1+\frac{1}{(p-1)p^s}\right). $$ This formula expresses a non-trivial divisor sum involving the Möbius function $μ$ and Euler's totient function $\varphi$ as a simple and explicit multiplicative expression. This is a generalization of the remarkable Dineva formula, which corresponds to $s=0$ and gives $n/\varphi(n)$ on the right-hand side. We explain why only squarefree divisors are involved, show how multiplicativity naturally comes into play, and interpret the identity as a finite Euler product. Beyond this one-parameter family of generalizations, we describe a general method for constructing similar formulas and present several examples. Finally, we reformulate these identities in terms of partial zeta functions, thus emphasizing their close relationship with the classical theory of Euler products and the Riemann zeta function. The connection with the Selberg sieve is briefly outlined.