On the $L$-series of F. Pellarin

TL;DR

This paper extends Pellarin's work on special values of certain $L$-series in finite characteristic, showing they can be analytically continued and interpolated at finite primes, revealing new structural insights.

Contribution

It demonstrates the analytic continuation and interpolation of Pellarin's $L$-series at finite primes, expanding understanding of their properties in finite characteristic.

Findings

$L$-series can be analytically continued with trivial zeroes.

The $L$-series can be interpolated at finite primes.

New structural insights into $L$-series in finite characteristic.

Abstract

The calculation, by L.\ Euler, of the values at positive even integers of the Riemann zeta function, in terms of powers of $\pi$ and rational numbers, was a watershed event in the history of number theory and classical analysis. Since then many important analogs involving $L$-values and periods have been obtained. In analysis in finite characteristic, a version of Euler's result was given by L.\ Carlitz \cite{ca2} in the 1930's which involved the period of a rank 1 Drinfeld module (the Carlitz module) in place of $\pi$. In a very original work \cite{pe2}, F.\ Pellarin has quite recently established a "deformation" of Carlitz's result involving certain $L$-series and the deformation of the Carlitz period given in \cite{at1}. Pellarin works only with the values of this $L$-series at positive integral points. We show here how the techniques of \cite{go1} also allow these new $L$-series to be analytically continued -- with associated trivial zeroes -- and interpolated at finite primes.