Related papers: Twisted second moment of modular $L$-functions to …

We introduce a family of rank-one local systems in the category of twisted $\mathcal{D}$-modules on a certain subvariety isomorphic to ${\mathbb{G}_{\text{m}}}^2$ of the affine flag variety of $\text{SL}_2$. We then give a criterion for…

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 the second moment is a sum over $t_j$ in a short interval. At the central point…

We consider negative moments of quadratic Dirichlet $L$--functions over function fields. Summing over monic square-free polynomials of degree $2g+1$ in $\mathbb{F}_q[x]$, we obtain an asymptotic formula for the $k^{\text{th}}$ shifted…

Assuming the Riemann Hypothesis, Soundararajan showed that $\displaystyle{\int_{0}^{T} \vert \zeta(1/2 + it)\vert^{2k} \ll T(\log T)^{k^2 + \epsilon}}$ . His method was used by Chandee to obtain upper bounds for shifted moments of the…

We improve the error term in the asymptotic formula for the twisted fourth moment of automorphic L functions of prime level and weight two proved by Kowalski, Michel and Vanderkam. As a consequence, we obtain a new subconvexity bound in the…

We use a relative trace formula on GL(2) to compute a sum of twisted modular L-functions anywhere in the critical strip, weighted by a Fourier coefficient and a Hecke eigenvalue. When the weight k or level N is sufficiently large, the sum…

We prove an asymptotic formula for the second moment of a product of two Dirichlet L-functions on the critical line, which has a power saving in the error term and which is uniform with respect to the involved Dirichlet characters. As…

We establish sharp upper bounds on shifted moments of quadratic Dirichlet $L$-functions over function fields. As an application, we prove some bounds for moments of quadratic Dirichlet character sums over function fields.

We establish power-saving estimates for general bilinear forms with Kloosterman sums modulo arbitrary q, including when both variables are shorter than the Polya-Vinogradov range. As an application, we obtain power-saving asymptotics for…

We prove hybrid subconvexity bounds twisted L-functions $L(s,f\times \chi)$ at the central point using a fourth moment estimate, including a new instance of the Burgess subconvexity bound.

We prove an asymptotic formula for the twisted first moment of Maass form symmetric square L-functions on the critical line and at the critical point. The error term is estimated uniformly with respect to all parameters.

Given $c,$ a positive integer, we give an explicit formula and an asymptotic formula for \[ \sum\chi(c)|L(1,\,\chi)|^{2}, \] where $\chi$ is the non-trivial Dirichlet character mod $f$ with $f>c.$

We prove hybrid subconvexity bounds for a wide class of twisted L-functions $L(s,f\times \chi)$ at the central point, including a new instance of the Weyl subconvexity bound.

We employ a regularized relative trace formula to establish a second moment estimate for twisted $L$-functions across all aspects over a number field. Our results yield hybrid subconvex bounds for both Hecke $L$-functions and twisted…

We analyse a collection of twisted mixed moments of the Riemann zeta function and establish the validity of asymptotic formulae comprising on some instances secondary terms of the shape $P(\log T) T^{C}$ for a suitable constant $C<1$ and a…

We prove that the central values of additive twists of a cuspidal $L$-function define a quantum modular form in the sense of Zagier, generalizing recent results of Bettin and Drappeau. From this we deduce a reciprocity law for the twisted…

We give a new proof of Heath-Brown's full asymptotic expansion for the second moment of Dirichlet L-functions and we obtain a corresponding asymptotic expansion for a twisted first moment of Hecke-Maass L-functions.

We compute an asymptotic formula for the mixed second moment of the $\mu$-th and $\nu$-th derivatives of quadratic Dirichlet $L$-functions over monic, irreducible polynomials in the function field setting.

We establish sharp upper bounds for shifted moments of quadratic Dirichlet $L$-function under the generalized Riemann hypothesis. Our result is then used to prove bounds for moments of quadratic Dirichlet character sums.

We prove an asymptotic formula for the second moment (up to height $T$) of the Riemann zeta function with two shifts. The case we deal with is where the real parts of the shifts are very close to zero and the imaginary parts can grow up to…