Special values of derivatives of certain L-functions

2024/08/25 by Tapas Chatterjee, Chatterjee, Tapas, Sonika Dhillon +1
Mathematics · #11J81 #11J86 #11J91 #11M06 #Analytic Number Theory Research #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2408.13737

openalex publication_date 2024/08/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Period polynomials, derivatives of L-functions, and zeros of polynomials
• On fractional derivatives and primitives of periodic functions
• Zeroes of L-series in characteristic p
• Non-real zeros of derivatives
• Derivatives of L-functions