On transcendence of non-periodic continued fractions associated with modular forms and arithmetic functions
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910021951946752 |
|---|---|
| author | Chatterjee, Tapas Mandal, Sagar |
| author_facet | Chatterjee, Tapas Mandal, Sagar |
| contents | The purpose of this article is two-folds. Firstly, we establish two sufficient conditions under which the sequence $\{f(n)\pmod{m}: n\geq1\}$ is non-periodic, where $f(n)$ is an arithmetic function. As consequences, we deduce that the sequences associated with the Ramanujan tau function $τ(n)$ as well as the Fourier coefficients of certain normalized Eisenstein series $E_k(z)$ modulo $m$ are non-periodic. Further, we deduce that the sequence arising from Nathanson's totient function $Φ(n)$, the classical Euler's totient function $φ(n)$, sum of divisor function $σ(n)$, their Dirichlet convolution $σ*φ(n)$, Jordan's totient function $J_k(n)$, and unitary totient function $φ^*(n)$ modulo $m$, are non-periodic for certain modulo $m$. In addition, we extend a result of Ayad and Kihel \cite{r1} on the non-periodicity of certain arithmetic function $g(n)$. On the other hand, we construct several transcendental numbers arising from the continued fractions attached with $τ(n)$, $E_k(z)$, $Φ(n)$, $g(n)$, $φ(n)$, $σ(n)$, $σ*φ(n)$, $J_k(n)$, and $φ^*(n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13300 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On transcendence of non-periodic continued fractions associated with modular forms and arithmetic functions Chatterjee, Tapas Mandal, Sagar General Mathematics Primary: 11A25, 11F11, 40A15, 11J81, Secondary: 11A55, 11B50 The purpose of this article is two-folds. Firstly, we establish two sufficient conditions under which the sequence $\{f(n)\pmod{m}: n\geq1\}$ is non-periodic, where $f(n)$ is an arithmetic function. As consequences, we deduce that the sequences associated with the Ramanujan tau function $τ(n)$ as well as the Fourier coefficients of certain normalized Eisenstein series $E_k(z)$ modulo $m$ are non-periodic. Further, we deduce that the sequence arising from Nathanson's totient function $Φ(n)$, the classical Euler's totient function $φ(n)$, sum of divisor function $σ(n)$, their Dirichlet convolution $σ*φ(n)$, Jordan's totient function $J_k(n)$, and unitary totient function $φ^*(n)$ modulo $m$, are non-periodic for certain modulo $m$. In addition, we extend a result of Ayad and Kihel \cite{r1} on the non-periodicity of certain arithmetic function $g(n)$. On the other hand, we construct several transcendental numbers arising from the continued fractions attached with $τ(n)$, $E_k(z)$, $Φ(n)$, $g(n)$, $φ(n)$, $σ(n)$, $σ*φ(n)$, $J_k(n)$, and $φ^*(n)$. |
| title | On transcendence of non-periodic continued fractions associated with modular forms and arithmetic functions |
| topic | General Mathematics Primary: 11A25, 11F11, 40A15, 11J81, Secondary: 11A55, 11B50 |
| url | https://arxiv.org/abs/2602.13300 |