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

Assuming the Riemann hypothesis, we investigate the distribution of gaps between the zeros of \xi'(s). We prove that a positive proportion of gaps are less than 0.796 times the average spacing and, in the other direction, a positive…

The main purpose of this note is to understand the arithmetic encoded in the special value of the $p$-adic $L$-function $\mathcal{L}_p^g(\mathbf{f},\mathbf{g},\mathbf{h})$ associated to a triple of modular forms $(f,g,h)$ of weights…

In the present paper, we show that, for every $\delta>0$, the function $(\log {\mathcal{L}}(s))^{(m)}$, where $m\in {\mathbb{N}} \cup \{ 0\}$ and ${\mathcal{L}} (s) := \sum_{n=1}^\infty a(n) n^{-s}$ is an element of the Selberg class…

Let $f$ and $g$ be two holomorphic or Hecke-Maass primitive cusp forms for $SL(2,\mathbb{Z})$ and $\chi$ be a primitive Dirichlet character of modulus $p$, an odd prime. A subconvex bound for the central values of the Rankin-Selberg…

We prove an explicit integral representation -- involving the pullback of a suitable Siegel Eisenstein series -- for the twisted standard $L$-function associated to a holomorphic vector-valued Siegel cusp form of degree $n$ and arbitrary…

We study the nonvanishing of twists of automorphic L-functions at the centre of the critical strip. Given a primitive character \chi modulo D satisfying some technical conditions, we prove that the twisted L-functions L(f.\chi,s) do not…

In this paper, we establish an asymptotic formula for the twisted second moment of $L$-functions associated with Hecke--Maass cusp forms for $\rm SL(3,\mathbb{Z})$, and further deduce a weighted zero-density estimate for these $L$-functions…

We establish sharp bounds for the second moment of symmetric-square $L$-functions attached to Hecke Maass cusp forms $u_j$ with spectral parameter $t_j$, where the second moment is a sum over $t_j$ in a short interval. At the central point…

We prove that there is a positive proportion of $L$-functions associated to cubic characters over $\mathbb{F}_q[T]$ that do not vanish at the critical point $s=1/2$. This is achieved by computing the first mollified moment using techniques…

From a spectral identity we obtain asymptotics with error term for the second integral moments of families of automorphic L-functions for GL(2) over an arbitrary number field according to twists by idele characters with arbitrary…

Let $N$ be a fixed positive integer, and let $f\in S_k(N)$ be a primitive cusp form given by the Fourier expansion $f(z)=\sum_{n=1}^{\infty} \lambda_f(n)n^{\frac{k-1}{2}}e(nz)$. We consider the partial sum $S(x,f)=\sum_{n\leq…

In this note we investigate the behavior at the central point of the symmetric square $L$-functions, the most frequently used $\rm{GL}(3)$ $L$-functions. We establish an asymptotic formula with arbitrary power saving for the first moment of…

In this paper, we study the second moment for $GL(2)\times GL(2)$ $L$-functions $L(\frac{1}{2},f\times g)$, which leads to a uniform subconvexity bound in the spectral aspect. In particular, if either $f$ or $g$ is a dihedral Maass newform,…

Let $\pi$ be a Hecke-Maass cusp form for $\rm SL_3(\mathbf{Z})$ and let $g$ be a holomorphic or Maass cusp form for $\rm SL_2(\mathbf{Z})$. Let $\chi$ be a primitive Dirichlet character of modulus $M=M_1M_2$ with $M_i$ prime, $i=1,2$.…

We will introduce two new classes of Dirichlet series which are monoids under multiplication. The first class $\mathfrak{A}^{\#}$ contains both the extended Selberg class $\mathscr{S}^{\#}$ of Kaczorowski and Perelli as well as many…

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…

Let $S_k$ be the space of holomorphic cusp forms of weight $k$ with respect to $SL_2(\mathbb{Z})$. Let $f \in S_k$ be a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. In this paper we consider the…

Generalizing our previous work on ``toroidal averages'', we study the average of special values of $L$-functions of the form $L(1/2,\chi^a)L(1/2,\chi^b)L(1/2,\chi^c)$ for integers $a$, $b$ and $c$, where $\chi$ varies over Dirichlet…

We introduce a new technique of completion for 1-cohomology which parallels the corresponding technique in the theory of mock modular forms. This technique is applied in the context of non-critical values of L-functions of GL(2,Q) cusp…

In this paper we calculate the asymptotics of the second moment of the Bessel periods associated to certain holomorphic cuspidal representations $(\pi, \pi')$ of $U(2,1) \times U(1,1)$ of regular infinity type (averaged over $\pi$). Using…