Abstract:
Let χ(modq), q>1, be a primitive Dirichlet character. We first present a detailed account of Linnik's deduction of the functional equation of L(s,χ) from the functional equation of ζ(s). Then we show that the opposite deduction can be obtained by a suitable modification of the method, involving finer arithmetic arguments.
This research was partially supported by Istituto Nazionale di Alta Matematica, by grant PRIN2015 Number Theory and Arithmetic Geometry and by grant no. 2013/11/B/ST1/02799 Analytic Methods in Arithmetic of the National Science Centre.
Citation:
J. Kaczorowski, A. Perelli, “A note on Linnik's approach to the Dirichlet L-functions”, Analytic and combinatorial number theory, Collected papers. On the occasion of the 125th anniversary of the birth of Academician Ivan Matveevich Vinogradov, Trudy Mat. Inst. Steklova, 296, MAIK Nauka/Interperiodica, Moscow, 2017, 123–132; Proc. Steklov Inst. Math., 296 (2017), 115–124
\Bibitem{KacPer17}
\by J.~Kaczorowski, A.~Perelli
\paper A note on Linnik's approach to the Dirichlet $L$-functions
\inbook Analytic and combinatorial number theory
\bookinfo Collected papers. On the occasion of the 125th anniversary of the birth of Academician Ivan Matveevich Vinogradov
\serial Trudy Mat. Inst. Steklova
\yr 2017
\vol 296
\pages 123--132
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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\crossref{https://doi.org/10.1134/S0371968517010095}
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\jour Proc. Steklov Inst. Math.
\yr 2017
\vol 296
\pages 115--124
\crossref{https://doi.org/10.1134/S0081543817010096}
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Linking options:
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https://doi.org/10.1134/S0371968517010095
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This publication is cited in the following 1 articles:
J. Kaczorowski, A. Perelli, “On the standard twist of the l-functions of half-integral weight cusp forms”, Nagoya Math. J., 240 (2020), 150–180