Related papers: Heegner points and Eisenstein series
We prove an asymptotic formula with a power saving error term for the (pure or mixed) second moment of central values of L-functions of any two (possibly equal) fixed cusp forms f, g twisted by all primitive characters modulo q, valid for…
In this paper, over imaginary quadratic fields, we consider the family of $L$-functions $L (s, f)$ for an orthonormal basis of spherical Hecke--Maass forms $f$ with Archimedean parameter $t_f$. We establish asymptotic formulae for the…
We study the second moment of the central values of quadratic twists of a modular $L$-function. Unconditionally, we obtain a lower bound which matches the conjectured asymptotic formula, while on GRH we prove the asymptotic formula itself.
We calculate certain "wide moments" of central values of Rankin--Selberg $L$-functions $L(\pi\otimes \Omega, 1/2)$ where $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_2$ over $\mathbb{Q}$ and $\Omega$ is a Hecke character…
We compute the second moment in the family of quadratic Dirichlet $L$-functions with prime conductors over $\mathbb{F}_q[x]$ when the degree of the discriminant goes to infinity, obtaining one of the lower order terms. We also obtain an…
We compute the second moment of the Dedekind zeta function of a quadratic field times an arbitrary Dirichlet polynomial of length $T^{1/11-\epsilon}$.
We propose a refined version of the existing conjectural asymptotic formula for the moments of the family of quadratic Dirichlet L-functions over rational function fields. Our prediction is motivated by two natural conjectures that provide…
In this paper we extend the hybrid Euler-Hadamard product model for quadratic Dirichlet $L$-functions associated to irreducible polynomials over function fields. We also establish an asymptotic formula for the first twisted moment in this…
We obtain a second moment formula for the L-series of holomorphic cusp forms, averaged over twists by Dirichlet characters modulo a fixed conductor Q. The estimate obtained has no restrictions on Q, with an error term that has a close to…
In this paper, we provide an alternative proof of Chandee and Li's result on the second moment of $\mathrm{GL}_4 \times \mathrm{GL}_2$ special $L$-values. Our method is conceptually more direct as it neither detects the…
Following a strategy suggested by Michel--Venkatesh, we study the cubic moment of automorphic $L$-functions on $\operatorname{PGL}_2$ using regularized diagonal periods of products of Eisenstein series. Our main innovation is to produce…
We calculate the first and second moments of L-functions in the family of quadratic twists of a fixed elliptic curve E over F_q[x], asymptotically in the limit as the degree of the twists tends to infinity. We also compute moments involving…
We compute the second moment of a certain family of Rankin-Selberg $L$-functions L(f x g, 1/2) where f and g are Hecke-Maass cusp forms on GL(n). Our bound is as strong as the Lindel\"of hypothesis on average, and recovers individually the…
We compute the second moment of the Riemann zeta function for shifted arguments over a domain that extends the ones in the literature. We use the Riemann-Siegel formula for the error term in the approximate functional equation and take the…
We obtain asymptotic formulas for the second and third moment of quadratic Dirichlet $L$--functions at the critical point, in the function field setting. We fix the ground field $\mathbb{F}_q$, and assume for simplicity that $q$ is a prime…
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 prove an asymptotic formula for the second moment of the first derivative of quadratic twists of modular $L$-functions with three leading order main terms. It improves the previous result of Kumar et al. with the first main term. The…
Recently, Keating and the second author of this paper devised a heuristic for predicting asymptotic formulas for moments of the Riemann zeta-function $\zeta(s)$. Their approach indicates how lower twisted moments of $\zeta(s)$ may be used…
We explicitly calculate the moments t_n of general Heisenberg Hamiltonians up to sixth order. They have the form of finite sums of products of two factors, the first factor being represented by a multigraph and the second factor being a…
We derive formulas for the terms in the conjectured asymptotic expansions of the moments, at the central point, of quadratic Dirichlet $L$-functions, $L(1/2,\chi_d)$, and also of the $L$-functions associated to quadratic twists of an…