Related papers: Moments of L-functions and Liouville-Green method

We study the $2k$-th moment of central values of the family of Dirichlet $L$-functions to a fixed prime modulus. We establish sharp lower bounds for all real $k \geq 0$ and sharp upper bounds for $k$ in the range $0 \leq k \leq 1$.

Higher Green functions are real-valued functions of two variables on the upper half plane which are bi-invariant under the action of a congruence subgroup, have logarithmic singularity along the diagonal, but instead of the usual equation…

Modifying a method of Jutila, we prove a t aspect subconvexity estimate for L-functions associated to primitive holomorphic cusp forms of arbitrary level that is of comparable strength to Good's bound for the full modular group, thus…

Given a large, square-free, smooth conductor, we establish the non-vanishing of the central values for at least $35.9\%$ of the primitive Dirichlet $L$-functions.

In this paper, we establish the gradient and Pogorelov estimates for $k$-convex-monotone solutions to parabolic $k$-Hessian equations of the form $-u_t\sigma_k(\lambda(D^2u))=\psi(x,t,u)$. We also apply such estimates to obtain a Liouville…

Let $f$ be a newform of prime level $p$ with any central character $\chi\, (\bmod\, p)$, and let $g$ be a fixed cusp form or Eisenstein series for $\hbox{SL}_{2}(\mathbb{Z})$. We prove the subconvexity bound: for any $\varepsilon>0$,…

Let f be a cusp form for SL(3, Z) associated with a generalized principal series representation of minimal weight d, spectral parameter r and associated L-function L(s, f). For $r \asymp d \asymp T$ the subconvexity bound $L(1/2, f) \ll…

Under the generalized Riemann Hypothesis (GRH), Baluyot, Chandee, and Li nearly doubled the range in which the density of low lying zeros predicted by Katz and Sarnak is known to hold for a large family of automorphic $L$-functions with…

Iwaniec and Sarnak showed that at the minimum 25% of L-values associated to holomorphic newforms of fixed even integral weight and level $N \rightarrow \infty$ do not vanish at the critical point when N is square-free and $\phi(N)\sim N$.…

In this paper, we establish the expected order of magnitude of the $k$th-moment of central values of the family of Dirichlet $L$-functions to a fixed prime modulus over function fields for all real $k \geq 0$.

We establish a sub-convexity estimate for Rankin-Selberg $L$-functions in the combined level aspect, using the circle method. If $p$ and $q$ are distinct prime numbers, $f$ and $g$ are non-exceptional newforms (modular or Maass) for the…

Let $g$ denote a fixed holomorphic Hecke cusp form of weight $k \equiv 0 \pmod{4}$ on $\mathrm{SL}_2(\mathbb{Z})$, and let $\pi$ be a fixed cuspidal automorphic representation of $\mathrm{GL}_3$. In this paper, we establish an asymptotic…

We investigate the large weight (k --> oo) limiting statistics for the low lying zeros of a GL(4) and a GL(6) family of L-functions, {L(s,phi x f): f in H_k(1)} and {L(s,phi times sym^2 f): f in H_k(1)}; here phi is a fixed even Hecke-Maass…

A well known result of Iwaniec and Sarnak states that for at least one third of the primitive Dirichlet characters to a large modulus q, the associated L-functions do not vanish at the central point. When q is a large power of a fixed…

In this paper, we prove the conjectured order lower bound for the $k$-th moment of central values of quadratic twisted self-dual $\textrm{GL}(3)$ $L$-functions for all $k\geq 1$, based on our recent work on the twisted first moment of…

We study the $1$-level density of low-lying zeros of quadratic Dirichlet $L$-functions by applying the $L$-functions Ratios Conjecture. We observe a transition in the main term as was predicted by the Katz-Sarnak heuristic as well as in the…

While Random Matrix Theory has successfully modeled many quantities of families of L-functions, it frequently cannot see the family's arithmetic. In some situations this requires an extended theory that inserts arithmetic factors depending…

We prove a Liouville theorem for ancient solutions to the supercritical Fujita equation \[\partial_tu-\Delta u=|u|^{p-1}u, \quad -\infty <t<0, \quad p>\frac{n+2}{n-2},\] which says if $u$ is close to the ODE solution…

The Katz-Sarnak Density Conjecture states that zeros of families of $L$-functions are well-modeled by eigenvalues of random matrix ensembles. For suitably restricted test functions, this correspondence yields upper bounds for the families'…

We study $\mathbf L^\infty$ entropy solutions to $2\times 2$ systems of conservation laws. We show that, if a uniformly convex entropy exists, these solutions satisfy a pair of kinetic equations (nonlocal in velocity), which are then shown…