On transcendence of non-periodic continued fractions associated with modular forms and arithmetic functions

TL;DR

This paper establishes conditions for non-periodicity of sequences derived from arithmetic functions modulo m and constructs transcendental numbers from continued fractions linked to these functions.

Contribution

It introduces new criteria for non-periodicity of various arithmetic sequences modulo m and constructs transcendental numbers from related continued fractions.

Findings

Sequences from Ramanujan tau and Eisenstein series are non-periodic modulo m.

Sequences from totient, divisor, and Jordan functions are non-periodic for certain m.

Constructs transcendental numbers from continued fractions associated with these functions.

Abstract

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 $\tau(n)$ 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 $\Phi(n)$, the classical Euler's totient function $\varphi(n)$, sum of divisor function $\sigma(n)$, their Dirichlet convolution $\sigma*\varphi(n)$, Jordan's totient function $J_k(n)$, and unitary totient function $\varphi^*(n)$ 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 $\tau(n)$, $E_k(z)$, $\Phi(n)$, $g(n)$, $\varphi(n)$, $\sigma(n)$, $\sigma*\varphi(n)$, $J_k(n)$, and $\varphi^*(n)$.