相关论文: Comparing zeros of distinct Dirichlet L-functions

We initiate the study of certain families of $L$-functions attached to characters of subgroups of higher-rank tori, and of their average at the central point. In particular, we evaluate the average of the values…

In a recent work, we introduced \textit{LC-functions} $L(s,f)$, associated to a certain real-analytic function $f$ at $0$, extending the concept of the Hurwitz zeta function and its formula. In this paper, we establish the existence of a…

Let $L(s)=\sum_{n=1}^{+\infty}\dfrac{a(n)}{n^s}$ be a Dirichlet series were $a(n)$ is a bounded completely multiplicative function. We prove that if $L(s)$ extends to a holomorphic function on the open half space $\Re s >1-\delta$,…

Let $L(s,\chi)$ be the Dirichlet $L$-function associated to a non trivial primitive Dirichlet character $\chi$ defined $\bmod\ q$, where $q$ is an odd prime. In this paper we introduce a fast method to compute $\vert L(1,\chi) \vert$ using…

A q-analogue of the Riemann zeta function was studied in [Kaneko et al. 03] via a certain q-series of two variables. We introduce in a similar way a q-analogue of the Dirichlet L-functions and make a detailed study of them, including some…

It is proved that \[ \sum_{\chi \bmod q}N(\sigma , T, \chi) \lesssim_{\epsilon} (qT)^{7(1-\sigma)/3+\epsilon}, \] where $N(\sigma, T, \chi)$ denote the number of zeros $\rho = \beta + it$ of $L(s, \chi)$ in the rectangle $\sigma \leq \beta…

We study the conjecture that $\sum_{n\leq x} \chi(n)=o(x)$ for any primitive Dirichlet character $\chi \pmod q$ with $x\geq q^\epsilon$, which is known to be true if the Riemann Hypothesis holds for $L(s,\chi)$. We show that it holds under…

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 $q$.…

As we have shown several years ago [Y2], zeros of $L(s, \Delta )$ and $L^(2)(s, \Delta )$ can be calculated quite efficiently by a certain experimental method. Here $\Delta$ denotes the cusp form of weight 12 with respect to SL$(2, Z)$ and…

We study the Iwasawa $\lambda$-invariant of Dirichlet characters $\chi$ of arbitrary order for odd primes $p$. From special values of the $p$-adic $L$-function and its derivative we derive several novel and easily computable criteria to…

The bivariate series $\theta (q,x):=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j$ defines a {\em partial theta function}. For fixed $q$, $\theta (q,.)$ is an entire function. We show that for $|q|\leq 0.108$ the function $\theta (q,.)$ has no…

In this paper, we use the Weyl-bound for Dirichlet $L$-functions to derive zero-density estimates for $L$-functions associated to families of fixed-order Dirichlet characters. The results improve on previous bounds given by the author when…

For any irreducible character $\chi$ of a finite group $G$, let $\theta(\chi)$ denote the proportion of elements $g\in G$ for which $\chi(g)$ is either zero or a root of unity. Then for any $L\in[1/2,1]$ and any $\epsilon>0$, there exists…

In recent years a variant of the resonance method was developed which allowed to obtain improved $\Omega$-results for the Riemann zeta function along vertical lines in the critical strip. In the present paper we show how this method can be…

An infinite class of relations between modular forms is constructed that generalizes evaluations of the Dirichlet beta function at odd positive integers. The work is motivated by a base case appearing in Ramanujan's Notebooks and a parallel…

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…

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.

Let $L(s,\chi)$ be the Dirichlet $L$-function associated to a non-principal primitive Dirichlet character $\chi$ defined modulo $q$, where $q\ge 3$. We prove, under the assumption of the Generalised Riemann Hypothesis, the validity of…

Let $M=M_1 M_2 M_3$ be the product of three distinct primes and let $\chi=\chi_1 \chi_2 \chi_3$ be a Dirichlet character of modulus $M$ such that each $\chi_i$ is a primitive character modulo $M_i$ for $i=1,2,3$. In this paper, we provide a…

Zeros of the Riemann zeta function and its derivatives have been studied by many mathematicians. Among, the number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been…