Nonvanishing of automorphic $L$-functions at the center of the critical strip

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 16, Tome 263 (2000), pp. 193-204

Résumé

Let $S_k(\Gamma_0(N)\chi)$ be the space of holomorphic $\Gamma_0(N)$-cusp forms of integral weight $k$ and of character $\chi(\operatorname{mod}n)$, let $f(z)$ be a newform of the space $S_k(\Gamma_0(N),\chi)$, and let $L_f(s)$ be the corresponding $L$-function. The following statements are proved. (1) Let $\mathscr F_0$ be the set of all newforms of $S_k(\Gamma_0(p),1)$, let $p$ be prime, and let $k\ge2$ be a constant even number. Then $$ \sum_{f\in\mathscr F_0:L_f(k/2)\ne0}1\gg\frac p{\log^2p} \quad (p\to\infty). $$ (2) Let $\mathscr F$ be the set of all Hecke eigenforms of the space $S_k(\Gamma_0(1),1)$ and let $k\equiv0\pmod 4$. Then $$ \sum_{f:\mathscr F_0:L_f(k/2)\ne0}1\gg\frac k{log^2k} \quad (k\to1). $$