On the L-series of F. Pellarin

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• 作者：David Goss ^{goss@math.ohio-state.edu}
• 关键词：Characteristic p L-series ; Pellarin L-series ; Bernoulli&ndash ; Carlitz numbers ; Drinfeld modules ; Special polynomials ; Trivial zeroes
• 刊名：Journal of Number Theory
• 出版年：2013
• 出版时间：March, 2013
• 年：2013
• 卷：133
• 期：3
• 页码：955-962
• 全文大小：160 K

文摘

The calculation, by L. Euler, of the values at positive even integers of the Riemann zeta function, in terms of powers of ¦Ð 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 (1937) , (1940) in the 1930s which involved the period of a rank 1 Drinfeld module (the Carlitz module) in place of ¦Ð. In a very original work (Pellarin, 2011 ), 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 Anderson and Thakur (1990) . Pellarin works only with the values of this L-series at positive integral points. We show here how the techniques of Goss (1996) also allow these new L-series to be analytically continued - with associated trivial zeroes - and interpolated at finite primes.