On Pellarin's $L$-series

Details On Pellarin's $L$-series On Pellarin's $L$-series

2011-12-29

arxiv.org/abs/1201.0030v2

Mathematics

Number Theory

Scientific paper

Necessary and sufficient conditions are given for a negative integer to be a \emph{trivial zero} of a new type of $L$-series recently discovered by F. Pellarin, and it is shown that any such trivial zero is simple. We establish the logarithmic growth of the degrees of the special polynomials associated to Pellarin's $L$-series in a wide set of circumstances. To do so the theory of Carlitz polynomial approximations is developed further for both additive and $\mathbb{F}_q$-linear functions. Finally a connection is made between the Wagner representation for $\chi_t$ and Pellarin's $L$-series.

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