Related papers: Monotonicity Results for Dirichlet L-functions

We present a simple analytic proof that L-functions of real non-principal Dirichlet characters are nonzero at 1.

In this paper, we estimate the proportion of zeros of Dirichlet $L$-functions on the critical line. Using Feng's mollifier and an asymptotic formula for the mean square of Dirichlet $L$-functions, we prove that averaged over primitive…

In this note we define L-functions of finite graphs and study the particular case of finite cycles in the spirit of a previous paper that studied spectral zeta functions of graphs. The main result is a suggestive equivalence between an…

We describe in detail three distinct families of generalized zeta functions built over the (nontrivial) zeros of a rather general arithmetic zeta or L-function, extending the scope of two earlier works that treated the Riemann zeros only.…

In this note our aim is to deduce some new monotonicity properties for a special combination of Bessel functions of the first kind by using a recently developed Mittag-Leffler expansion for the derivative of a normalized Bessel function of…

We compute asymptotic formulae for the mollified first and second moments for the family of quadratic Dirichlet $L$-functions in the function field setting. As an application, we obtain non-vanishing results for the derivatives of the…

In this paper, we study the number of additional zeros of Dirichlet $L$-function caused by multiplicity by using Asymptotic Large Sieve. Then in asymptotic terms we prove that there are more than 80.124% of zeros of the family of Dirichlet…

We develop a discrete spectral framework for Dirichlet $L$-functions that reveals a combinatorial structure underlying their special values and connects this to their zeros. Our approach approximates the classical Dirichlet series by finite…

Let $L(s,\chi)$ be a fixed Dirichlet $L$-function. Given a vertical arithmetic progression of $T$ points on the line $\Re(s)=1/2$, we show that $\gg T \log T$ of them are not zeros of $L(s,\chi)$. This result provides some theoretical…

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…

In this paper our aim is to present the completely monotonicity and convexity properties for the Wright function. As consequences of these results, we present some functional inequalities. Moreover, we derive the monotonicity and…

The main purpose of this paper is to establish bounds on the second moment of $L\big(\tfrac{1}{2}+it,\chi\big)$, averaged over families of fixed order characters. A discrete version of the main result is also stated, from which zero-density…

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 study the family of Dirichlet $L$-functions of all even primitive characters of conductor at most $Q$, where $Q$ is a parameter tending to $\infty$. For an arbitrary positive integer $k$, we approximate the twisted $2k$th moment of this…

In a recent paper with Sprang and Zudilin, the following result was proved: if $a$ is large enough in terms of $\varepsilon>0$, then at least $2^{(1-\varepsilon)\frac{\log a}{\log \log a}}$ values of the Riemann zeta function at odd…

We simplify the proof of some widely used theoretical theorems, extending their applicability, while correcting some erroneous results. We also generalize key results and present new results that contribute to the development of the theory.…

In this paper, we first investigate the monotonicity and limit problem of the fractional integral functions. By fixed point theorem and these new results of the fractional integral functions, we present that the Riemann-Liouville fractional…

We estimate the $1$-level density of low-lying zeros of $L(s,\chi)$ with $\chi$ ranging over primitive Dirichlet characters of conductor $\in [Q/2,Q]$ and for test functions whose Fourier transform is supported in $[- 2 - 50/1093, 2 +…

We consider a sum of the derivatives of Dirichlet $L$-functions over the zeros of Dirichlet $L$-functions. We give an asymptotic formula for the sum.

For any $\theta>\frac13$, we show that there are constants $c_1,c_2>0$ that depend only on $\theta$ for which the following property holds. If $\chi_1,\chi_2$ are two distinct primitive Dirichlet characters modulo $q$, and $T\ge…