Generalized relations between arithmetic functions
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917214510120960 |
|---|---|
| author | Pain, Jean-Christophe |
| author_facet | Pain, Jean-Christophe |
| contents | The aim of this article is to present in a self-contained way identities arising in elementary number theory, among which the following one: $$ \sum_{d\mid n}\frac{μ^2(d)}{φ(d)\,d^s}=\prod_{p\mid n}\left(1+\frac{1}{(p-1)p^s}\right). $$ This formula expresses a non-trivial divisor sum involving the Möbius function $μ$ and Euler's totient function $φ$ as a simple and explicit multiplicative expression. This is a generalization of the remarkable Dineva formula, which corresponds to $s=0$ and gives $n/φ(n)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_14521 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generalized relations between arithmetic functions Pain, Jean-Christophe Number Theory The aim of this article is to present in a self-contained way identities arising in elementary number theory, among which the following one: $$ \sum_{d\mid n}\frac{μ^2(d)}{φ(d)\,d^s}=\prod_{p\mid n}\left(1+\frac{1}{(p-1)p^s}\right). $$ This formula expresses a non-trivial divisor sum involving the Möbius function $μ$ and Euler's totient function $φ$ as a simple and explicit multiplicative expression. This is a generalization of the remarkable Dineva formula, which corresponds to $s=0$ and gives $n/φ(n)$ 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. |
| title | Generalized relations between arithmetic functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2601.14521 |