Related papers: Zero free regions for Dirichlet series

We prove joint universality theorems on the half plane of absolute convergence for general classes of Dirichlet series with an Euler-product, where in addition to vertical shifts we also allow scaling. This generalizes our recent joint…

We obtain a necessary and sufficient condition in order that a semi-plane of the form $\Re(s)>r$, $r\in \mathbb{R}$, is free of zeros of a given Dirichlet polynomial. The result may be considered a natural generalization of a well-known…

For a wide class of Dirichlet series associated to automorphic forms, we show that those without Euler products must have zeros within the region of absolute convergence. For instance, we prove that if f is a classical holomorphic modular…

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…

We prove that the Riemann zeta-function $\zeta(\sigma + it)$ has no zeros in the region $\sigma \geq 1 - 1/(55.241(\log|t|)^{2/3} (\log\log |t|)^{1/3})$ for $|t|\geq 3$. In addition, we improve the constant in the classical zero-free…

We give a new proof that the Riemann zeta function is nonzero in the half-plane $\{s\in{\mathbb C}:\sigma>1\}$. A novel feature of this proof is that it makes no use of the Euler product for $\zeta(s)$.

The purpose of this paper is twofold. One is to investigate the properties of the zeros of cross-products of Bessel functions or derivatives of ultraspherical Bessel functions, as well as the properties of the zeros of the derivative of the…

We define generalized Li coefficients, called $\tau-$Li coefficients for a very broad class $\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)$ of $L-$functions that contains the Selberg class, the class of all automorphic $L-$functions and…

We show that for a positive proportion of real primitive Dirichlet characters chi, the associated Dirichlet L-function L(s,chi) has no zeros on the positive real axis. Prior to this it was not known whether or not there were infinitely many…

We prove two unconditional upper bounds on the gaps between ordinates of consecutive non-trivial zeros of a general $L$-function $L(s)$. This extends previous work of Hall and Hayman (2000) on the Riemann zeta-function and work of Siegel…

We prove a new explicit zero-free region for the Riemann zeta-function, drawing substantially on Heath-Brown's seminal work on Linnik's constant. Using these ideas we are able to prove that $\zeta(\sigma + it)\ne 0$ whenever $t\geq 3$ and…

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…

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.

Let $\chi$ be a real and non-principal Dirichlet character, $L(s,\chi)$ its Dirichlet $L$-function and let $p$ be a generic prime number. We prove the following result: If for some $0\leq \sigma<1$ the partial sums $\sum_{p\leq…

Let $F$ be a linear combination of $N\geq 1$ Dirichlet $L$-functions attached to even (or odd) primitive characters with the same modulus. Selberg proved that a positive proportion of non-trivial zeros of $F$ lie on the critical line. Our…

Let $K$ be a number field and, for an integral ideal $\mathfrak{q}$ of $K$, let $\chi$ be a character of the narrow ray class group modulo $\mathfrak{q}$. We establish various new and improved explicit results, with effective dependence on…

A relevant result independently obtained by Rayleigh and Sneddon on an identity on series involving the zeros of Bessel functions of the first kind is derived by an alternative method based on Laplace transforms. Our method leads to a…

In this note we investigate connections between zero density estimates for the Riemann zeta function and large value estimates for Dirichlet polynomials. It is well known that estimates of the latter type imply estimates of the former type.…

Building on the work of K. Chiba (J. Algebra 263 (2003), 75-87), we present sufficient conditions for the completion of a division ring with respect to the metric defined by a discrete valuation function to contain a free field, i.e. the…

We give explicit and extended versions of some of Siegel's results. We extend the validity of Siegel's asymptotic development in the second quadrant to most of the third quadrant. We also give precise bounds of the error; this allows us to…