Under certain mild analytic assumptions one obtains a lower bound, essentially of order $r$, for the number of zeros and poles of a Dirichlet series in a disk of radius $r$. A more precise result is also obtained under more restrictive assumptions but still applying to a large class of Dirichlet series.

Introduction. In 1974, N. Levinson showed that at least 1/3 of the zeros of the Riemann ζ-function are on the critical line ([19]). Today it is known (Conrey, [6]) that at least 40.77% of the zeros of ζ(s) are on the critical line and at least 40.1% are on the critical line and are simple. In [16] and [17], Hilano showed that Levinson's original result is also valid for Dirichlet L-series. This paper is a shortened version of parts of the dissertation [3], the full details of...

For $p$ an odd prime, we show that the Fekete polynomial $f_p\left(t\right)={\sum }_{a=0}^{p-1}\left(\frac{a}{p}\right)t^a$ has $\sim {\kappa }_{0}p$ zeros on the unit circle, where $0.500813\>{\kappa }_0\>0.500668$. Here ${\kappa }_0-1/2$ is the probability that the function $1/x+1/(1-x)+{\sum }_{n\in \mathbb{Z}:n\ne 0,1}{\delta }_n/(x-n)$ has a zero in $]0,1[$, where each ${\delta }_n$ is $\pm 1$ with y $1/2$. In fact $f_p\left(t\right)$ has absolute value $\sqrt{p}$ at each primitive $p$th root of unity, and we show that if $|f_p\left(e\right(2i\pi (K+\tau )/p\left)\right)|\<ϵ\sqrt{p}$ for some $\tau \in ]0,1[$ then there is a zero of $f$ close to this arc.

A family of Zeta functions built as Dirichlet series over the Riemann zeros are shown to have meromorphic extensions in the whole complex plane, for which numerous analytical features (the polar structures, plus countably many special values) are explicitly displayed.

This work is about a generalization of Kœcher’s zeta function. Let $V$ be an Euclidean simple Jordan algebra of dimension $n$ and rank $m$, $E$ an Euclidean space of dimension $N$, $\varphi$ a regular self-adjoint representation of $V$ in $E$, $Q$ the quadratic form associated to $\varphi$, $\Omega$ the symmetric cone associated to $V$ and $G\left(\Omega \right)$ its automorphism group$$G\left(\Omega \right)=\{g\in GL(V\left)\right|g\left(\Omega \right)=\Omega \}.$$($H_1$) Assume that $V$ and $E$ have $\mathbf{Q}$-structures $V_{\mathbf{Q}}$ and $E_{\mathbf{Q}}$ respectively and $\varphi$ is defined over $\mathbf{Q}$. Let $L$ be a lattice in $E_{\mathbf{Q}}$. The zeta series associated to $\varphi$ and $L$ is defined by$${\zeta }_L\left(s\right)=\sum _{l\in {\Gamma }_{\circ }\setminus L^{\text{'}}}\left[\det \right(Q\left(l\right)){]}^{-s},\forall s\in \mathbf{C}$$where $L^{\text{'}}=\{l\in L|\det \left(Q\right(l\left)\right)\ne 0\}$,...