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On the \(L\)-series of F. Pellarin. (English) Zbl 1281.11045

Summary: 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, Duke Math. J. 3, 503–517 (1937; Zbl 0017.19501); ibid. 7, 62–67 (1940; Zbl 0024.24410)] in the 1930s which involved the period of a rank 1 Drinfeld module (the Carlitz module) in place of \(\pi \). In a very original work [Ann. Math. (2) 176, No. 3, 2055–2093 (2012; Zbl 1336.11064)], 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 [G. W. Anderson and D. S. Thakur in [Ann. Math. (2) 132, No. 1, 159–191 (1990; Zbl 0713.11082)]. Pellarin works only with the values of this \(L\)-series at positive integral points. We show here how the techniques of the author [Basic structures of function field arithmetic. Berlin: Springer (1996; Zbl 0874.11004)] also allow these new \(L\)-series to be analytically continued – with associated trivial zeros – and interpolated at finite primes.

MSC:

11F52 Modular forms associated to Drinfel’d modules
11C08 Polynomials in number theory
11B68 Bernoulli and Euler numbers and polynomials
11M38 Zeta and \(L\)-functions in characteristic \(p\)
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References:

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