Chennai Mathematical Institute


On non-vanishing and linear independence of special values of Dirichlet series

Abhishek Bharadwaj
Chennai Mathematical Institute.


We study the linear independence and non-vanishing results concerning the special values of Dirichlet series. For an arithmetic function ${f : \mathbb{N} \to \mathbb{C}}$, we have the corresponding Dirichlet series, $ {L(s,f) = \sum_{n \ge 1} f(n)n^{-s}}$. Typical examples are the Riemann zeta function and the Dirichlet $L$ function for a character $\chi$, denoted by $L(s,\chi)$. The non-vanishing of the special values of these $L$ functions encode some arithmetic information. For instance, the proof of the infinitude of primes in arithmetic progressions relies on the non-vanishing of $L(1,\chi)$ for non-trivial characters $\chi$.

In the first part of the talk, we study the relations between the special values of these functions over arbitrary number fields. We also study the relations of Euler constants in the real and $p$-adic setup.

In the second part of this talk, we focus on a Conjecture of Erd\H{o}s concerning the non-vanishing of $L(1,f)$ for certain periodic functions $f$ with restricted values. We discuss some cases associated with this conjecture.