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St. Petersburg Mathematical Journal

This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.

ISSN 1547-7371 (online) ISSN 1061-0022 (print)

The 2020 MCQ for St. Petersburg Mathematical Journal is 0.68.

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On the distribution of values of $L(1,\mathrm {sym}^2f)$
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by O. M. Fomenko
Translated by: G. V. Kuz′mina and O. M. Fomenko
St. Petersburg Math. J. 17 (2006), 1031-1046
DOI: https://doi.org/10.1090/S1061-0022-06-00939-3
Published electronically: September 20, 2006

Abstract:

Let $S_k(\mathrm {SL}(2,\mathbb {Z}))^+$ be the set of holomorphic Hecke eigencuspforms $f$ of weight $k$ with respect to $\mathrm {SL}(2,\mathbb {Z})$. Let $L(s,\mathrm {sym}^2f)$ be the symmetric square of the Hecke L-function of a cusp form $f$. The moments of $L(1,\mathrm {sym}^2f)$, $f\in S_k(\mathrm {SL}(2,\mathbb {Z}))^+$, are computed for a pure imaginary order. The limiting distribution of $\log L(1,\mathrm {sym}^2 f)$, $f\in S_k(\mathrm {SL}(2,\mathbb {Z}))^+$, is studied in the weight aspect. Namely, the limiting distribution function, the limiting characteristic function and its Euler product are investigated, and the rate of convergence of frequencies to the limiting distribution is measured.

As a consequence, new facts on the limiting distribution of $\mathrm {SL}(1,\mathrm {sym}^2f)$ are obtained not only in the case of the holomorphic Hecke eigencuspforms $f$, but also in the case of the Hecke–Maass eigencuspforms $f$.

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Bibliographic 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
  • Received by editor(s): March 10, 2005
  • Published electronically: September 20, 2006
  • Additional Notes: Partially supported by RFBR (grant no. 05-01-00930).

  • Dedicated: Dedicated to Yuriĭ Vladimirovich Linnik’s 90th birthday anniversary
  • © Copyright 2006 American Mathematical Society
  • Journal: St. Petersburg Math. J. 17 (2006), 1031-1046
  • MSC (2000): Primary 11M41
  • DOI: https://doi.org/10.1090/S1061-0022-06-00939-3
  • MathSciNet review: 2202450