Related papers: Gaps between zeros of GL(2) $L$-functions

This is a reworked version of the paper. An idea that allows us to circumvent limitations of previous approaches is not to apply arithmetic-geometric mean inequality and the second moment asymptotics to the entire segment $[1/2-a/\log…

The principal aim of this article is to attach and study $p$-adic $L$-functions to cohomological cuspidal automorphic representations $\Pi$ of $\mathrm{GL}(2n)$ over a totally real field $F$ admitting a Shalika model. We use a modular…

Let $F$ be a self-dual Hecke-Maa\ss\ form for ${\rm{GL}}(3)$ underlying the symmetric square lift of a ${\rm{GL}}(2)$-newform of square-free level and trivial nebentypus. In this paper, we are interested in the first moments of the central…

We prove a strong form of the trivial zero conjecture at the central point for the $p$-adic $L$-function of a non-critically refined self-dual cohomological cuspidal automorphic representation of $\mathrm{GL}_2$ over a totally real field,…

Let $1\le N<M$ with $N$ and $M$ coprime and square-free. Through classical analytic methods we estimate the first moment of central $L$-values $ L(1/2,f\times g) $ where $f\in S^*_k(N)$ runs over primitive holomorphic forms of level $N$ and…

Let $M$ be a squarefree positive integer and $P$ a prime number coprime to $M$ such that $P\sim M^\eta$ with $0 < \eta < 2/5$. We simplify the proof of subconvexity bounds for $L(\frac{1}{2},f\otimes\chi)$ when $f$ is a primitive…

We provide an explicit integral representation for L-functions of pairs (F,g) where F is a holomorphic genus 2 Siegel newform and g a holomorphic elliptic newform, both of squarefree levels and of equal weights. When F,g have level one,…

We show that for any sufficiently large $T,$ there exists a subinterval of $[T,2T]$ of length at least $2.766 \times \frac{2\pi}{\log{T}},$ in which the function $t \mapsto \zeta(1/2 + it)$ has no zeros.

We study the restriction of the Bump-Friedberg integrals to affine lines $\{(s+\alpha,2s),s\in\C\}$. It has a simple theory, very close to that of the Asai $L$-function. It is an integral representation of the product…

We establish the equivalence of conjectures concerning the pair correlation of zeros of $L$-functions in the Selberg class and the variances of sums of a related class of arithmetic functions over primes in short intervals. This extends the…

We obtain (conditional and unconditional) results on large values of $L$-functions $L(s,\chi)$ in the critical strip $1/2 \leq \Re s \leq 1$ when the character $\chi$ runs through a thin subgroup of all characters modulo an integer $q$.…

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…

Let $\Pi$ be a cohomological cuspidal automorphic representation of ${\rm GL}_{2n}(\mathbb A)$ over a totally real number field $F$. Suppose that $\Pi$ has a Shalika model. We define a rational structure on the Shalika model of $\Pi_f$.…

We give a conjecture for the moments of the Dedekind zeta function of a Galois extension via the hybrid product method. The moments of the product of primes are evaluated using the Montgomery-Vaughan mean value theorem whilst for the…

Let $f$ be a holomorphic cusp form for $SL_2(\mathbb{Z})$ of weight $k>1$. In these notes, we follow Munshi to prove the Burgess bound $$ L(1/2+it,f)\ll_{f,\varepsilon} (1+|t|)^{1/2-1/8+\varepsilon}. $$

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…

In this note, we prove that given a Hecke-Maass cusp form $f$ for $SL_2(\mathbb{Z})$ and a sufficiently large integer $q=q_1q_2$ with $q_j\asymp \sqrt{q}$ being prime numbers for $j=1,2$, there exists a primitive Dirichlet character $\chi$…

Let $q \in \mathbb{Z} [i]$ be prime and $\chi $ be the primitive quadratic Hecke character modulo $q$. Let $\pi$ be a self-dual Hecke automorphic cusp form for $\mathrm{SL}_3 (\mathbb{Z} [i] )$ and $f$ be a Hecke cusp form for $\Gamma_0 (q)…

Let $\pi$ be a Hecke cusp form for $\mathrm{SL}_3(\mathbb{Z})$. We bound the second moment average of $L(s,\pi)$ over a short interval to obtain the subconvexity estimate $$ L(1/2+it, \pi) \ll_{\pi, \varepsilon}…

We consider the family of Rankin-Selberg convolution L-functions of a fixed SL(3, Z) Maass form with the family of Hecke-Maass cusp forms on SL(2, Z). We estimate the second moment of this family of L-functions with a "long" integration in…