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On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve

Published:2013-08-10
Printed: Jun 2014
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, POLAND
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Abstract

We study analytic properties function $m(z, E)$, which is defined on the upper half-plane as an integral from the shifted $L$-function of an elliptic curve. We show that $m(z, E)$ analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for $m(z, E)$ in the strip $|\Im{z}|\lt 2\pi$.
 Keywords: L-function, Möbius function, explicit formulae, elliptic curve
 MSC Classifications: 11M36 - Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas 11G40 - $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]