Related papers: The Zero Density Theorem for the Rankin-Selberg L-…

In this article we show simultaneous non-vanishing of two Rankin-Selberg $L$-functions by proving an asymptotic result in weight aspect. The main input of this paper is to remove the $t$-integral dependence from the result of Blomer-Harcos…

A generalized Riemann hypothesis states that all zeros of the completed Hecke $L$-function $L^*(f,s)$ of a normalized Hecke eigenform $f$ on the full modular group should lie on the vertical line $Re(s)=\frac{k}{2}.$ It was shown by Kohnen…

We establish effective versions of Oppenheim's conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed shift vectors and generic quadratic forms. When the shift is rational we prove a counting result which…

We introduce a notion of Hecke-monicity for functions on certain moduli spaces associated to torsors of finite groups over elliptic curves, and show that it implies strong invariance properties under linear fractional transformations.…

Using the mollifier method, we show that for a positive proportion of holomorphic Hecke eigenforms of level one and weight bounded by a large enough constant, the associated symmetric square $L$-function does not vanish at the central point…

A density-functional theory is established for inhomogeneous superfluids at finite temperature, subject to time-dependent external fields in isothermal conditions. After outlining parallelisms between a neutral superfluid and a charged…

We address the problem of identifying a Hecke-Maass cusp form $f$ of full level from the central values of the Rankin-Selberg $L$-functions $L(1/2,f\otimes h)$ where $h$ runs through the set of Hecke-Maass eigenforms of full level. We prove…

The theory of Selberg zeta functions is generalized to higher rank spaces. Applications towards analytic torsion numbers are given.

Let $\mathfrak{F}_n$ be the set of unitary cuspidal automorphic representations of $\mathrm{GL}_n$ over a number field $F$, and let $S\subseteq\mathfrak{F}_n$ be an arbitrary finite subset. Given $\pi_0\in\mathfrak{F}_{n_0}$, we establish…

We study the local statistics of zeros of $L$-functions attached to Artin-Scheier curves over finite fields. We consider three families of Artin-Schreier $L$-functions: the ordinary, polynomial (the $p$-rank 0 stratum) and odd-polynomial…

In this paper we apply to the zeros of families of $L$-functions with orthogonal or symplectic symmetry the method that Conrey and Snaith used to calculate the $n$-correlation of the zeros of the Riemann zeta function. This method uses the…

Poincare-type series, such as Selberg's, are known to produce automorphic functions, in the hyperbolic half-plane, the decompositions of which into eigenfunctions (genuine or generalized) of the automorphic Laplacian contain all modular…

In this note we investigate connections between zero density estimates for the Riemann zeta function and large value estimates for Dirichlet polynomials. It is well known that estimates of the latter type imply estimates of the former type.…

In this paper, we use the Weyl-bound for Dirichlet $L$-functions to derive zero-density estimates for $L$-functions associated to families of fixed-order Dirichlet characters. The results improve on previous bounds given by the author when…

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…

The Rankin-Selberg method for studying Langlands' automorphic $L$-functions is to find integral representations, involving certain Fourier coefficients of cusp forms and Eisenstein series, for these functions. In this thesis we develop the…

We derive explicit formulae for the subalgebra zeta functions of all higher Heisenberg Lie algebras over an arbitrary compact discrete valuation ring $\mathfrak{o}$. To this end, we develop Hecke-theoretic techniques for the enumeration, by…

The thesis gave a fine study on the distribution of the coefficients of automorphic L-functions for GL(m) with m>1. In particular we have treated two types of problems: change of signs of these coefficients (when they are real) and their…

In the paper we prove an explicit formula for the central values of certain Rankin L-functions. These L-functions are L-functions of Hilbert newforms over a totally real field F, twisted by unitary Hecke characters of a totally imaginary…

The principles of density-functional theory are studied for finite lattice systems represented by graphs. Surprisingly, the fundamental Hohenberg-Kohn theorem is found void in general, while many insights into the topological structure of…