Related papers: Non-vanishing of Dirichlet L-functions at the cent…

In this paper, for a given Dirichlet character mod $N$ with $4\nmid N$, we give a lower bound of order $\sqrt{s/\log(s)}$ for the dimension of the $\mathbb{Q}(e^{2i\pi/N})$-vector space spanned by the values of its $L$-function at integers…

We prove a non-vanishing result for central values of $L$-functions on GL(3), by using the mollification method and the Kuznetsov trace formula.

We prove an asymptotic for the eighth moment of Dirichlet $L$-functions averaged over primitive characters $\chi$ modulo $q$, over all moduli $q\leq Q$ and with a short average on the critical line, conditionally on GRH. We derive the…

In this paper, we exhibit upper and lower bounds with explicit constants for some objects related to entire $L$-functions in the critical strip, under the generalized Riemann hypothesis. The examples include the entire Dirichlet…

In this paper, we consider the family $\{L_j(s)\}_{j=1}^{\infty}$ of $L$-functions associated to an orthonormal basis $\{u_j\}_{j=1}^{\infty}$ of even Hecke-Maass forms for the modular group $SL(2, Z)$ with eigenvalues…

Let $q$ be a positive integer ($\geq 2$), $\chi$ be a Dirichlet character modulo $q$, $L(s, \chi)$ be the attached Dirichlet $L$-function, and let $L^\prime(s, \chi)$ denote its derivative with respect to the complex variable $s$. Let $t_0$…

For a fairly general family of L-functions, we survey the known consequences of the existence of asymptotic formulas with power-sawing error term for the (twisted) first and second moments of the central values in the family. We then…

We survey a number of different methods for computing $L(\chi,1-k)$ for a Dirichlet character $\chi$, with particular emphasis on quadratic characters. The main conclusion is that when $k$ is not too large (for instance $k\le100$) the best…

We obtain a lower bound on the number of quadratic Dirichlet L-functions over the rational function field which vanish at the central point $s = 1/2$. This is in contrast with the situation over the rational numbers, where a conjecture of…

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 study the 2k-th power moment of Dirichlet L-functions L(s,\chi) at the centre of the critical strip (s=1/2), where the average is over all primitive characters \chi (mod q). We extend to this case the hybrid Euler-Hadamard product…

Let $\chi$ denote a primitive, non-quadratic Dirichlet character with conductor $q$, and let $L(s, \chi)$ denote its associated Dirichlet $L$-function. We show that $|L(1, \chi)| \geq 1/(9.12255 \log(q/\pi))$ for sufficiently large $q$, and…

We prove a subconvexity bound for the central value L(1/2, chi) of a Dirichlet L-function of a character chi to a prime power modulus q=p^n of the form L(1/2, chi)\ll p^r * q^(theta+epsilon) with a fixed r and theta\approx 0.1645 < 1/6,…

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…

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…

Let $\chi$ be a non-principal Dirichlet character and $L(s, \chi)$ be the associated Dirichlet $L$-function. Let us use $\mathcal{L}(s,\chi)$ to denote its logarithmic derivative $L'(s, \chi)/L(s, \chi)$. We first prove some arithmetic…

In the present paper, we study large values of Dirichlet $L$- functions inside the critical strip. For every $1/2<\sigma<1$, we show that for $q$ sufficiently large, there exists a non-principal character $\chi$ modulo $q$ and a constant…

Let $f$ be a fixed holomorphic primitive cusp form of even weight $k$, level $r$ and trivial nebentypus $\chi_r$. Let $q$ be an odd prime with $(q,r)=1$ and let $\chi$ be a primitive Dirichlet character modulus $q$ with $\chi\neq\chi_r$. In…

In this paper, we establish a new lower bound for the number of low-lying zeros of Dirichlet $L$-functions $L(s, \chi)$ on the critical line within extremely short intervals. Specifically, for a sufficiently large prime $P$ and real number…

Let $\chi$ a primitive character$\pmod q$ and consider the Dirichlet $L$-function $$L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^s}.$$ We give a new proof of an upper bound of Heath-Brown for $|L(s,\chi)|$ on the critical line $s=1/2+it$