Related papers: New developments toward the Gonek Conjecture on th…

The real and complex zeros of some special entire functions such as Wright, hyper-Bessel, and a special case of generalized hypergeometric functions are studied by using some classical results of Laguerre, Obreschkhoff, P\'olya and Runckel.…

In 2003, Garunk\v{s}tis provided a lower bound for the lower density of the universality theorem for the Riemann zeta-function. In this paper, we generalize this result for the hybrid joint universality theorem for Dirichlet $L$-functions…

It is proved that the Epstein zeta-function corresponding to a binary positive definite quadratic form with integer coefficients has a positive proportion of its non-trivial zeros on the critical line.

In the paper the well known Riemann Hypothesis is proven. The proof is based on uniform approximation of the zeta function discs of the critical strip placed to the right from the critical line.The basic moment is a use of a new mesure…

In this work we consider an equation for the Riemann zeta-function in the critical half-strip. With the help of this equation we prove that finding non-trivial zeros of the Riemann zeta-function outside the critical line would be equivalent…

In the present paper we give a simple mathematical foundation for describing the zeros of the Selberg zeta functions $Z_X$ for certain very symmetric infinite area surfaces $X$. For definiteness, we consider the case of three funneled…

By employing contour integration the derivation of a generalized double finite series involving the Hurwitz-Lerch zeta function is used to derive closed form formulae in terms of special functions. We use this procedure to find special…

It is well known that the Riemann zeta function, as well as several other $L$-functions, is universal in the strip $1/2<\sigma<1$; this is certainly not true for $\sigma>1$. Answering a question of Bombieri and Ghosh, we give a simple…

We prove that the partial zeta function introduced in [9] is a rational function, generalizing Dwork's rationality theorem.

Let $0 < a \le 1$, $s,z \in {\mathbb{C}}$ and $0 < |z|\le 1$. Then the Hurwitz-Lerch zeta function is defined by $\Phi (s,a,z) := \sum_{n=0}^\infty z^n(n+a)^{-s}$ when $\sigma :=\Re (s) >1$. In this paper, we show that the Hurwitz zeta…

The Riemann Hypothesis is a conjecture that all non-trivial zeros of Riemann Zeta function are located on the critical line in the complex plane. Hundreds of propositions in function theory and analytic number theory rely on this…

In this paper, we prove a new identity for values of the Hurwitz zeta function which contains as particular cases Koecher's identity for odd zeta values, the Bailey-Borwein-Bradley identity for even zeta values and many other interesting…

This paper show that two doubly infinite series generate a totally nonnegative Hurwitz-type matrix if and only if their ratio represents an S-functions of a certain kind. The doubly infinite case needs a specific approach, since the ratios…

In this note we evaluate multiple integrals that play a crucial role in the theory of irrationality of zeta function

Given a polynomial \[ f(x)=a_0x^n+a_1x^{n-1}+\cdots +a_n \] with positive coefficients $a_k$, and a positive integer $M\leq n$, we define a(n infinite) generalized Hurwitz matrix $H_M(f):=(a_{Mj-i})_{i,j}$. We prove that the polynomial…

Several arguments against the truth of the Riemann hypothesis are extensively discussed. These include the Lehmer phenomenon, the Davenport-Heilbronn zeta-function, large and mean values of $|\zeta(1/2+it)|$ on the critical line, and zeros…

We prove a monotonicity property of the Hurwitz zeta function which, in turn, translates into a chain of inequalities for polygamma functions of different orders. We provide a probabilistic interpretation of our result by exploiting a…

In this article, it is proved that the non-trivial zeros of the Riemann zeta function must lie on the critical line, known as the Riemann hypothesis.

The functional equation for Riemann's Zeta function is studied, from which it is shown why all of the non-trivial, full-zeros of the Zeta function $\zeta (s)$ will only occur on the critical line {$\sigma=1/2$} where {$s=\sigma+I \rho$},…

This is an anthology of series involving rational, factorial, and power functions expressed in terms of special functions. New finite expansions involving quotient functions expressed in terms of the Hurwitz-Lerch zeta function are given.…