A note on Dirichlet-like series attached to polynomials
Frédéric Chapoton
TL;DR
This work generalizes Dirichlet L-functions by attaching a periodic coefficient χ and a polynomial P to define Dirichlet-like series L_{χ,P}(s) = sum_{n≥1} χ(n) P'(n)/P(n)^s, and establishes holomorphic continuation to the whole complex plane with explicit negative-integer values given by L_{χ,P}(1−m) = − Ψ_χ(P^m)/m. It also extends the framework to L_{A,χ,P}(s) with a finite initial segment, yielding a corrected negative-integer formula that includes contributions from the first A−1 terms, and proves the results via a Taylor-type expansion of P obtained from its roots. The paper provides a detailed example with χ_3 and P = X(X+U), showing how the corresponding Ψ_χ leads to polynomials p_m(u) that exhibit congruence patterns modulo primes and suggest p-adic interpolation, indicating rich arithmetic structure beyond classical Dirichlet L-functions. These findings connect to Bernoulli-type linear forms Ψ_χ and offer a unified approach to a family of zeta- and L-function–like objects attached to polynomials.
Abstract
Some Dirichlet-like functions, attached to a pair (periodic function, polynomial) are introduced and studied. These functions generalize the standard Dirichlet L-functions of Dirichlet characters. They have similar properties, being holomorphic on thefull complex plane and having simple values on negative integers.