# On the transcendence of certain infinite series

## Meeting Details

Abstract: In 1737, Euler solved the Basel problem by evaluating the values of the Riemann zeta-function at even positive integers. He showed that $\zeta(2k)$ is a rational multiple of $\pi^{2k}$. Since then, there have been several generalizations of Euler's result. One such question is to evaluate and determine the arithmetic nature of the more general series, $\sum_{n=1}^{\infty} A(n) / B(n)$, where $A(X)$ and $B(X)$ are suitable polynomials. Although it is possible to express these sums in terms of the polygamma functions, their arithmetic nature remains a mystery. In this talk, we will discuss analogs of this problem in two distinct scenarios.