We establish sharp bounds for the second moment of symmetric-square $L$-functions attached to Hecke Maass cusp forms $u_j$ with spectral parameter $t_j$, where is a sum over $t_j$ in short interval. At central point $s=1/2$ $L$-function, our interval smaller than previous known results. More specifically, $|t_j|$ size $T$, $T^{1/5}$, while best was $T^{1/3}$ from work Lam. A little higher up on...

We show that the percentage of primitive forms level one and weight $4k\rightarrow \infty$ for which associated $L$-function at central point is no less than $(\log{k})^{-2}$ least 20%. The key ingredients our proof are Kuznetsov convolution formula Liouville-Green method.

In this paper, we establish the expected order of magnitude $k$th-moment quadratic Dirichlet $L$-functions associated to hyperelliptic curves genus $g$ as well prime moduli over a fixed finite field $\mathbb{F}_{q}$ for all real $k \geq 0$.

Abstract We study wide moments of Dirichlet L-functions using analytic properties the Lerch zeta function. Among other things we obtain an asymptotic expansion (with arbitrary twists) extending results Heath-Brown. also give applications to non-vanishing.