Refinements of Erdős's irrationality criterion for certain sparse infinite series
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908794149142528 |
|---|---|
| author | Kaneko, Hajime Suzuki, Yuta Tachiya, Yohei |
| author_facet | Kaneko, Hajime Suzuki, Yuta Tachiya, Yohei |
| contents | In this paper, we establish new irrationality criteria for certain sparse power series. As applications of these criteria, we generalize a result of Erdős and obtain several irrationality results for various infinite series involving the classical arithmetic functions. For example, we prove that for any integers $t\ge2$ and $k\geq0$, the numbers \[ \sum_{n=1}^{\infty} \frac{d(n)^k}{t^{σ(n)}} \quad\text{and}\quad \sum_{n=1}^{\infty} \frac{d(n)^k}{t^{ϕ(n)}} \] are both irrational, where $d(n)$, $σ(n)$, and $ϕ(n)$ denote the number of divisors, the sum of divisors, and Euler's totient functions, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20743 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Refinements of Erdős's irrationality criterion for certain sparse infinite series Kaneko, Hajime Suzuki, Yuta Tachiya, Yohei Number Theory Primary 11J72, Secondary 11J81, 11A25 In this paper, we establish new irrationality criteria for certain sparse power series. As applications of these criteria, we generalize a result of Erdős and obtain several irrationality results for various infinite series involving the classical arithmetic functions. For example, we prove that for any integers $t\ge2$ and $k\geq0$, the numbers \[ \sum_{n=1}^{\infty} \frac{d(n)^k}{t^{σ(n)}} \quad\text{and}\quad \sum_{n=1}^{\infty} \frac{d(n)^k}{t^{ϕ(n)}} \] are both irrational, where $d(n)$, $σ(n)$, and $ϕ(n)$ denote the number of divisors, the sum of divisors, and Euler's totient functions, respectively. |
| title | Refinements of Erdős's irrationality criterion for certain sparse infinite series |
| topic | Number Theory Primary 11J72, Secondary 11J81, 11A25 |
| url | https://arxiv.org/abs/2601.20743 |