The symmetric squares of Hecke $L$-funktions and Fourier coefficients of cusp forms

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 18, Tome 286 (2002), pp. 200-214

Résumé

Let $S_k(\Gamma_0(N))$ be the space of cusp forms of even weight $k$ for $\Gamma_0(N)$, let $\mathscr F_0$ be the set of all newforms in $S_k(\Gamma_0(N))$, and let $\mathscr H_2(s,f)$ be the symmetric square of the Hecke $L$-function of a form $f\in\mathscr F_0$. It is proved that for $N=p$ we have $$ \sum_{f\in\mathscr F_0,\mathscr H_2(1/2,f)\ne0}1\gg N^{1-\varepsilon}, $$ where the $\ll$-constant depends only on $\varepsilon$ and $k$. Let $f(z)\in S_k(\Gamma(N))$: $$ f(z)=\sum^{\infty}_{n=1}a_f(n)e^{2\pi inz}, \qquad a_f(n)n^{-(k-1)/2}=b_f(n). $$ The distribution of values of the sums $$ \sum_{n\le X}b_f(n) \quad\text{and}\quad \sum_{n\le X}b_f(n)^2 $$ for increasing $X$ and $N$ is studied.