Related papers: Non-vanishing for cubic $L$--functions

In this paper, we study central values of the family of quadratic twists of modular $L$-functions of moduli $8p$, with $p$ ranging over odd primes. Assuming the truth of the generalized Riemann hypothesis, we establish a positive proportion…

Let L(E/Q,s) be the L-function of an elliptic curve E defined over the rational field Q. We examine the vanishing and non-vanishing of the central values L(E,1,\chi) of the twisted L-function as \chi ranges over Dirichlet characters of…

In Iwaniec-Sarnak [IS] the percentages of nonvanishing of central values of families of GL_2 automorphic L-functions was investigated. In this paper we examine the distribution of zeros which are at or neat s=1/2 (that is the central point)…

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 study asymptotically the twisted second moment of the family of modular $L$-functions to a fixed modulus. As an application, we establish sharp lower bounds for all real $k \geq 0$ and sharp upper bounds for $k$ in the range $0 \leq k…

Let $F$ be a Hecke-Maa\ss\ cusp form for $\mathrm{SL}(3,\mathbb{Z})$. We obtain the first non-trivial upper bound of the second moment of $L(F,s)$ in $t$-aspect: $$\int_{T}^{2T}|L(F,1/2+it)|^2 dt\ll_{F,\varepsilon}…

Let $\pi$ be an irreducible cuspidal automorphic representation of a quasi-split unitary group ${\rm U}_{\mathfrak n}$ defined over a number field $F$. Under the assumption that $\pi$ has a generic global Arthur parameter, we establish the…

Let E be an elliptic curve and \rho an Artin representation, both defined over the rational numbers. Let p be a prime at which E has good reduction. We prove that there exists an infinite set of Dirichlet characters \chi, ramified only at…

Let $D$ be a definite quaternion algebra over $\mathbb{Q}$ and $\mathcal{O}$ an Eichler order in $D$ of square-free level. We study distribution of the toric periods of algebraic modular forms of level $\mathcal{O}$. We focus on two…

We investigate in this paper the vanishing at $s=1$ of the twisted $L$-functions of elliptic curves $E$ defined over the rational function field $\mathbb{F}_q(t)$ (where $\mathbb{F}_q$ is a finite field of $q$ elements and characteristic…

In a recent work arXiv:2004.14450, it has been shown that $L$-functions associated with arbitrary non-zero cusp forms take large values at the central critical point. The goal of this note is to derive analogous results for twists of…

We prove that more than nine percent of the central values $L(\frac{1}{2},\chi_p)$ are non-zero, where $p\equiv 1 \pmod{8}$ ranges over primes and $\chi_p$ is the real primitive Dirichlet character of conductor $p$. Previously, it was not…

We study the low-lying zeros of certain Artin $L$-functions associated with $D_4$-quartic function fields. Specifically, we prove that when ordered by conductor, at least $77\%$ of these $L$-functions are non-vanishing at the central point.…

We prove a non-vanishing result for families of $\GL_n\times\GL_n$ Rankin-Selberg $L$-functions in the critical strip, as one factor runs over twists by Hecke characters. As an application, we simplify the proof, due to Luo, Rudnick, and…

Meher et al. [Proc. Amer. Math. Soc. 147 (2019)] have recently established that $L-$functions attached to certain cusp forms of half-integral weight have infinitely many zeros on the critical line. Kim [J. Numb. Th. 253 (2023)] obtained…

For any periodic function $f:{\mathbb N} \to {\mathbb C}$ with period $q$, we study the Dirichlet series $L(s,f):=\sum_{n\geq 1} f(n)/n^s.$ It is well-known that this admits an analytic continuation to the entire complex plane except at…

We give a theorem on the effective non-vanishing problem for algebraic surfaces in positive characteristic. For the Kawamata-Viehweg vanishing, the logarithmic Kollar vanishing and the logarithmic semipositivity, we give their…

We prove an asymptotic formula with a power saving error term for the fourth moment of the family of Dirichlet $L$-functions to modulus $q$ mollified by a Dirichlet polynomial of length $q^{\frac1{22}-\ve}$, valid for all moduli…

Let $\chi$ range over the $(p-1)/2$ even Dirichlet characters modulo a prime $p$ and denote by $\theta (x,\chi)$ the associated theta series. The asymptotic behaviour of the second and fourth moments proved by Louboutin and the author…

We establish a priori $L^\infty$-estimates for non-negative solutions of a semilinear nonlocal Neumann problem. As a consequence of these estimates, we get non-existence of non-constant solutions under suitable assumptions on the diffusion…