Irrationality Measures for Continued Fractions With Arithmetic Functions

Jaroslav Hančl, Kalle Leppälä

Let $f(n)$ or the base-$2$ logarithm of $f(n)$ be either $d(n)$ (the divisor function), $\sigma(n)$ (the divisor-sum function), $\varphi(n)$ (the Euler totient function), $\omega(n)$ (the number of distinct prime factors of $n$) or $\Omega(n)$ (the total number of prime factors of $n$). We present good lower bounds for $\bigl|\frac MN-\alpha\bigr|$ in terms of $N$, where $\alpha=[0;f(1),f(2),\ldots]$.