Fractional moments of automorphic $L$-functions.~II

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 25, Tome 383 (2010), pp. 179-192

Résumé

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 $T\to\infty$ $$ \int^T_1\big|L(\sigma+it,\mathrm{sym}^2f)\big|^{2k}\,dt=C\cdot T+O\left(T^{1-(2\sigma-1)/\{2(3-2\sigma)\}+\varepsilon}\right), $$ where $k>0$ and $\frac12\sigma1$. We obtain an analogous result for $L(s,f)$ conditionally and the asymptotics $$ \int^T_1\big|L(\sigma+it,f)\big|^{2k}\,dt\sim C_1\cdot T,\qquad01, $$ unconditionally. Bibl. 11 titles.