Automorphic $L$-functions in the weight aspect

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 20, Tome 314 (2004), pp. 221-246

Résumé

Let $S_k(\Gamma)$ be the space of holomorphic $\Gamma$-cusp forms $f(z)$ of even weight $k\geqslant12$ for $\Gamma=SL(2,\mathbb Z)$, and let $S_k(\Gamma)^+$ be the set of all Hecke eigenforms from this space with the first Fourier coefficient $a_f(1)=1$. For $f\in S_k(\Gamma)+$, consider the Hecke $L$-function $L(s,f)$. Let $$ S(k\leqslant K)=\bigcup_{\substack{12\leqslant k\leqslant K\\k\text{ even}}}S_k(\Gamma)^+. $$ It is proved that for large $K$, $$ \sum_{f\in S(k\leqslant K)}L\Bigl(\frac12,f\Bigr)^4\ll K^{2+\varepsilon}, $$ where $\varepsilon>0$ is arbitrary. For $f\in S_k(\Gamma)^+$ let $L(s,\operatorname{sym}^2f)$ denote the symmetric square $L$-function. It is proved that as $k\to\infty$ the frequence $$ \frac{\#\{f\mid f\in S_k(\Gamma)^+,L(1,\operatorname{sym}^2f)\leqslant x\}}{\#\{f\mid f\in S_k(\Gamma)^+\}} $$ converges to a distribution function $G(x)$ at every point of continuity of the latter, and for the corresponding characteristic function an explicit expression is obtained.