Related papers: Note on the Zeros of a Dirichlet Function

The aim of this article is to investigate how various Riemann Hypotheses would follow only from properties of the prime numbers. To this end, we consider two classes of $L$-functions, namely, non-principal Dirichlet and those based on cusp…

We show that the existence of arithmetic progressions with few primes, with a quantitative bound on "few", implies the existence of larger gaps between primes less than x than is currently known unconditionally. In particular, we derive…

We study the Dirichlet series associated with the integers whose radix-$b$ representation misses certain (fixed) digits. The existence of a meromorphic continuation to the entire complex plane, which was already well-known as a general fact…

Given $k\in\mathbb N$, we study the vanishing of the Dirichlet series $$D_k(s,f):=\sum_{n\geq1} d_k(n)f(n)n^{-s}$$ at the point $s=1$, where $f$ is a periodic function modulo a prime $p$. We show that if $(k,p-1)=1$ or $(k,p-1)=2$ and…

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…

This article is devoted to deduce the expression of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional…

The Linear Independence hypothesis (LI), which states roughly that the imaginary parts of the critical zeros of Dirichlet L-functions are linearly independent over the rationals, is known to have interesting consequences in the study of…

We study non-vanishing of Dirichlet $L$-functions at the central point under the unlikely assumption that there exists an exceptional Dirichlet character. In particular we prove that if $\psi$ is a real primitive character modulo $D \in…

Assuming the generalized Riemann hypothesis, we rediscover and sharpen some of the best known results regarding the distribution of low-lying zeros of Dirichlet $L$-functions. This builds upon earlier work of Omar, which relies on the…

We consider the joint value distribution of Dirichlet $L$-functions in the critical strip $\frac{1}{2} < \sigma < 1$. We show that the values of distinct Dirichlet $L$-functions are dependent in the sense that they do not behave like…

In this paper, we first prove the existence of solutions to Dirichlet problems involving the fractional $g$-Laplacian operator and lower order terms by appealing to sub- and supersolution methods. Moreover, we also state the existence of…

The aim of this paper is to improve the upper bound for the exceptional zeroes $\beta_0$ of Dirichlet $L$-functions. We do this by improving on explicit estimate for $L'(\sigma, \chi)$ for $\sigma$ close to unity.

We prove some identities, which involve the non-trivial zeros of the Riemann zeta function. From them we derive some convergent asymptotic expansions related to the work by Cram\'er, and also new representations for some arithmetical…

For any $s \in \mathbb{C}$ with $\Re(s)>0$, denote by $\eta_{n-1}(s)$ the $(n-1)^{th}$ partial sum of the Dirichlet series for the eta function $\eta(s)=1-2^{-s}+3^{-s}-\cdots \;$, and by $R_n(s)$ the corresponding remainder. Denoting by…

We examine the behaviour of the zeros of the real and imaginary parts of $\xi(s)$ on the vertical line $\Re s = 1/2+\lambda$, for $\lambda \neq 0$. This can be rephrased in terms of studying the zeros of families of entire functions $A(s) =…

The two-point correlation function for the zeros of Dirichlet L-functions at a height E on the critical line is calculated heuristically using a generalization of the Hardy-Littlewood conjecture for pairs of primes in arithmetic…

A class theorem is presented and proved: the complex Fourier transforms of a certain class of exponential functions have all their zeros on the real line. A class of basis functions is first considered, and the class is then extended via…

A lower bound for the dimension of the $\Q$-vector space spanned by special values of a Dirichlet series with periodic coefficients is given. As a corollary, it is deduced that both special values at even integers and at odd integers…

We study three special Dirichlet series, two of them alternating, related to the Riemann zeta function. These series are shown to have extensions to the entire complex plane and we find their values at the negative integers (or residues at…

In this work, we obtain an asymptotic formula for the twisted mean square of a Dirichlet $L$-function with a longer mollifier, whose coefficients are also more general than before. As an application we obtain that, for every Dirichlet…