Special values of derivatives of certain $L$-functions

TL;DR

This paper investigates the non-vanishing and transcendence of derivatives of certain $L$-functions at zero, characterizing functions with zero or transcendental derivatives, and exploring algebraic independence under Schanuel's conjecture.

Contribution

It characterizes when derivatives of specific $L$-functions are zero or transcendental and provides new examples and conditional results on algebraic independence.

Findings

Identifies functions with zero or transcendental derivatives of $L$-functions.

Provides examples where derivatives are zero using cyclotomic units.

Establishes algebraic independence of derivatives' values under Schanuel's conjecture.

Abstract

In this paper we address the question of non-vanishing of $L'(0,f)$ where $f$ is an algebraic valued periodic function. In 2011, Gun, Murty and Rath studied the nature of special values of the derivatives of even Dirichlet-type functions and proved that it can be either zero or transcendental. Here for some special cases we characterize the set of functions for which $L'(0,f)$ is zero or transcendental. Using a theorem of Ramachandra about multiplicative independence of cyclotomic units we also provide some non-trivial examples of functions where $L'(0,f)$ is zero. Finally, assuming Schanuel's conjecture we derive the algebraic independence of special values of derivatives of $L$-functions.