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Mean value theorems for automorphic $ L$-functions


Author: O. M. Fomenko
Translated by: the author
Original publication: Algebra i Analiz, tom 19 (2007), nomer 5.
Journal: St. Petersburg Math. J. 19 (2008), 853-866
MSC (2000): Primary 11M41
Published electronically: June 27, 2008
MathSciNet review: 2381948
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ f$ be a holomorphic Hecke eigencuspform of even weight $ k\ge 12$ for $ \operatorname{SL}(2, \mathbb{Z})$ and let $ L(s, \operatorname{sym}^2f)$ be the symmetric square $ L$-function of $ f$. Let $ C(x)$ be the summatory function of the coefficients of $ L(s,\operatorname{sym}^2 f)$. The true order is found for

$\displaystyle \int^{x}_{0}C(y)^2\,dy. $


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Additional Information

O. M. Fomenko
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: fomenko@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-08-01024-8
Keywords: Symmetric square $L$-function, summatory function, Euler product, Voronoi formula, mean value
Received by editor(s): April 5, 2007
Published electronically: June 27, 2008
Dedicated: Dedicated to the 100th anniversary of D. K. Faddeev’s birth
Article copyright: © Copyright 2008 American Mathematical Society