On the L-series of F. Pellarin

The calculation, by L. Euler, of the values at positive even integers of the Riemann zeta function, in terms of powers of π 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 Eulerʼs result was given by L. Carlitz (1937) [Ca2], , (1940) [Ca3], in the 1930s which involved the period of a rank 1 Drinfeld module (the Carlitz module) in place of π. In a very original work (Pellarin, 2011 [Pe2], ), F. Pellarin has quite recently established a “deformation” of Carlitzʼs result involving certain L-series and the deformation of the Carlitz period given in Anderson and Thakur (1990) [AT1], . Pellarin works only with the values of this L-series at positive integral points. We show here how the techniques of Goss (1996) [Go1] also allow these new L-series to be analytically continued – with associated trivial zeroes – and interpolated at finite primes.