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Zap. Nauchn. Sem. POMI, 2010, Volume 383, Pages 179–192 (Mi znsl3880)  

This article is cited in 1 scientific paper (total in 1 paper)

Fractional moments of automorphic $L$-functions. II

O. M. Fomenko

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences, St. Petersburg, Russia

Abstract: Let $f(z)$ be a holomorphic Hecke eigencuspform of even weight $\varkappa\ge12$ for $\mathrm{SL}(2,\mathbb Z)$. We consider the automorphic $L$-functions $L(s,f)$ (Hecke's $L$-function of $f$) and $L(s,\mathrm{sym}^2f)$ (Shimura's symmetric square $L$-function of $f$). Under the Riemann hypothesis for $L(s,\mathrm{sym}^2f)$, we prove the following asymptotic formula as $T\to\infty$
$$ \int^T_1|L(\sigma+it,\mathrm{sym}^2f)|^{2k} dt=C\cdot T+O(T^{1-(2\sigma-1)/\{2(3-2\sigma)\}+\varepsilon}), $$
where $k>0$ and $\frac12<\sigma<1$.
We obtain an analogous result for $L(s,f)$ conditionally and the asymptotics
$$ \int^T_1|L(\sigma+it,f)|^{2k} dt\sim C_1\cdot T,\qquad0<k<1, $$
unconditionally. Bibl. 11 titles.

Key words and phrases: automorphic $L$-function, critical strip, fractional moment.

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English version:
Journal of Mathematical Sciences (New York), 2011, 178:2, 219–226

UDC: 511.466+517.863
Received: 26.04.2010

Citation: O. M. Fomenko, “Fractional moments of automorphic $L$-functions. II”, Analytical theory of numbers and theory of functions. Part 25, Zap. Nauchn. Sem. POMI, 383, POMI, St. Petersburg, 2010, 179–192; J. Math. Sci. (N. Y.), 178:2 (2011), 219–226

Citation in format AMSBIB
\Bibitem{Fom10}
\by O.~M.~Fomenko
\paper Fractional moments of automorphic $L$-functions.~II
\inbook Analytical theory of numbers and theory of functions. Part~25
\serial Zap. Nauchn. Sem. POMI
\yr 2010
\vol 383
\pages 179--192
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl3880}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2011
\vol 178
\issue 2
\pages 219--226
\crossref{https://doi.org/10.1007/s10958-011-0541-1}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-80053525751}


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    This publication is cited in the following articles:
    1. O. M. Fomenko, “Extreme values of automorphic $L$-functions”, J. Math. Sci. (N. Y.), 193:1 (2013), 136–144  mathnet  crossref  mathscinet
  • Записки научных семинаров ПОМИ
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