Generalized relations between arithmetic functions

TL;DR

This paper presents a family of identities linking divisor sums involving the Möbius and Euler totient functions to explicit multiplicative formulas, generalizing known results and connecting to zeta functions and sieve theory.

Contribution

It introduces a new one-parameter family of identities in number theory, generalizing the Dineva formula and providing a method to construct similar formulas with applications to zeta functions.

Findings

Derived a generalized divisor sum identity involving μ and φ functions.

Expressed identities as finite Euler products and related them to partial zeta functions.

Connected the identities to classical Euler product theory and the Selberg sieve.

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: $$ \sum_{d\mid n}\frac{\mu^2(d)}{\varphi(d)\,d^s}=\prod_{p\mid n}\left(1+\frac{1}{(p-1)p^s}\right). $$ This formula expresses a non-trivial divisor sum involving the M\"obius function $\mu$ 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.