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

The purpose of this article is two-folds. Firstly, we establish two sufficient conditions under which the sequence $\{f(n)(\mathrm{m}\mathrm{o}\mathrm{d}m):n\ge 1\}$ 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 $\mathrm{\Phi }(n)$, the classical Euler’s totient function $\phi (n)$, sum of divisor function $\sigma (n)$, their Dirichlet convolution $\sigma \ast \phi (n)$, Jordan’s totient function $J_k(n)$, and unitary totient function ${\phi }^{\ast }(n)$ 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 $\tau (n)$, $E_k(z)$, $\mathrm{\Phi }(n)$, $g(n)$, $\phi (n)$, $\sigma (n)$, $\sigma \ast \phi (n)$, $J_k(n)$, and ${\phi }^{\ast }(n)$.