Related papers: Real zeros of quadratic Dirichlet L-functions

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…

In this paper, we compute and verify the positivity of the Li coefficients for the Dirichlet $L$-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125 (2007) no.1, 50-58; J. Number Theory 130 (2010)…

Let C(q,+) be the set of even, primitive Dirichlet characters (mod q). Using the mollifier method we show that L^{(k)}(1/2,chi) is not equal to zero for almost all the characters chi in C(q,+) when k and q are large. Here, L^{(k)}(s,chi) is…

We show that for at least 3/8 of the primitive Dirichlet characters $\chi$ of large prime modulus, the central value $L(1/2,\chi)$ does not vanish.

We present some monotonicity results for Dirichlet $L$-functions associated to real primitive characters. We show in particular that these Dirichlet $L$-functions are far from being logarithmically completely monotonic. Also, we show that,…

Given a large, square-free, smooth conductor, we establish the non-vanishing of the central values for at least $35.9\%$ of the primitive Dirichlet $L$-functions.

We give an explicit upper bound for non-principal Dirichlet $L$-functions on the line $s=1+it$. This result can be applied to improve the error in the zero-counting formulae for these functions.

We establish an asymptotic formula for the first moment and derive an upper bound for the second moment of L-functions associated with the complete family of primitive cubic Dirichlet characters defined over the Eisenstein field. Our…

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 given a Hecke-Maass form $f$ for $\text{SL}(2, \mathbb{Z})$ and a sufficiently large prime $q$, there exists a primitive Dirichlet character $\chi$ of conductor $q$ such that the $L$-values $L(\tfrac{1}{2}, f \otimes \chi)$…

We prove asymptotics for mollified first and second moments of subfamilies of Dirichlet $L$-functions given by shrinking angular restrictions on the root number. Using these moments, we prove that for even primitive characters with prime…

The existence of non trivial zeros off the critical line for a function obtained by analytic continuation of a particular Dirichlet series is studied. Contrary to what has been presumed for a long time, we prove that such zeros cannot…

This text shows the existence of large (3.54 times the average) gaps between consecutive zeros, on the critical line, of some Dirichlet $L$-functions $L(s,\chi),$ with $\chi$ being an even primitive Dirichlet character.

In this paper, we prove some zero density theorems for certain families of Dirichlet $L$-functions. More specifically, the subjects of our interest are the collections of Dirichlet $L$-functions associated with characters to moduli from…

We consider Dirichlet $L$-functions $L(s, \chi)$ where $\chi$ is a real, non-principal character modulo $q$. Using Pintz's refinement of Page's theorem, we prove that for $q\geq 3$ the function $L(s, \chi)$ has at most one real zero $\beta$…

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 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 $\chi$…

We show that for at least $\frac{5}{13}$ of the primitive Dirichlet characters $\chi$ of large prime modulus, the central value $L(\frac{1}{2},\chi)$ does not vanish, improving on the previous best known result of $\frac{3}{8}$.

Given $c,$ a positive integer, we give an explicit formula and an asymptotic formula for \[ \sum\chi(c)|L(1,\,\chi)|^{2}, \] where $\chi$ is the non-trivial Dirichlet character mod $f$ with $f>c.$

We prove some new log-free density theorems for zeros of Dirichlet L-functions (which accordingly are more sharp than earlier ones near to the boundary line of the critical strip). The results can be applied in several problems of prime…