Related papers: A note on Hardy's theorem

We provide a lower bound for the dimension of the vector space spanned by 1 and by the values of the Riemann Zeta function at the first odd integers. As a consequence, the Zeta function takes infinitely many irrational values at odd…

We verify numerically, in a rigorous way using interval arithmetic, that the Riemann hypothesis is true up to height $3\cdot10^{12}$. That is, all zeroes $\beta + i\gamma$ of the Riemann zeta-function with $0<\gamma\leq 3\cdot 10^{12}$ have…

In Part I an odd meromorphic function f(s) has been constructed from the Riemann zeta-function evaluated at one-half plus s. The conjunction of the Riemann hypothesis and hypotheses advanced by the author in Part I is assumed. In Part IV we…

This analysis which uses new mathematical methods aims at proving the Riemann hypothesis and figuring out an approximate base for imaginary non-trivial zeros of zeta function at very large numbers, in order to determine the path that those…

We study the values taken by the Riemann zeta-function $\zeta$ on discrete sets. We show that infinite vertical arithmetic progressions are uniquely determined by the values of $\zeta$ taken on this set. Moreover, we prove a joint discrete…

A proof for the original Riemann hypothesis is proposed based on the infinite Hadamard product representation for the Riemann zeta function and later generalized to Dirichlet L-functions. The extension of the hypothesis to other functions…

A discussion involving the evaluation of the sum $$\sum_{T<\g\le T+H}|\zeta(1/2+i\gamma)|^2$$ and some related integrals is presented, where $\gamma\,(>0)$ denotes imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. It…

In this work we show that the Riemann hypothesis for the Dedekind zeta--function $\zeta_{\mathrm{K}}(s)$ of an algebraic number field $\mathrm{K}$ is equivalent to a problem of the rate of convergence of certain discrete measures defined…

For a certain function $J(s)$ we prove that the identity $$\frac{\zeta(2s)}{\zeta(s)}-\left(s-\frac{1}{2}\right)J(s)=\frac{\zeta(2s+1)}{\zeta(s+1/2)}, $$ holds in the half-plane Re$(s)>1/2$ and both sides of the equality are analytic in…

In this paper we provide a proof of the Riemann Hypothesis by relating the non-trivial zeros of the zeta function to a certain Sturm-Liouville eigenvalue problem on a finite interval.

Let $\gamma$ denote the imaginary parts of complex zeros $\rho = \beta+i\gamma$ of $\zeta(s)$. The problem of analytic continuation of the function $G(s) := \sum\limits_{\gamma > 0}\gamma^{-s}$ to the left of the line $\Re s = -1$ is…

The secondary zeta function is defined as a generalized zeta series over the imaginary parts of non-trivial zeros assuming (RH). This function admits Laurent series expansion at the double pole at $s=1$. In this article, we derive a new…

In this paper we provide a new series representation for the values of Riemann zeta function at integer arguments, namely: $ \zeta(m)=\sum_{n=1}^{\infty}\frac{m(-1)^{n-1}\Gamma(1-\omega_{m}n)...\Gamma(1-\omega_{m}^{m-1}n)}{n!n^m}$, where…

The aim of this paper is to give a full detail of the proof given by Harder of a theorem on the denominator of the Eisenstein class for $\mathrm{SL}_2(\mathbb{Z})$ and to show that the theorem has some interesting applications including the…

We give a new equivalent condition for the Riemann hypothesis consisting in an order condition for certain finite rational combinations of the values of the Riemann zeta-function at even positive integers.

In this note, we prove the existence of infinitely many zeros of certain generalized Hurwitz zeta functions in the domain of absolute convergence. This is a generalization of a classical problem of Davenport, Heilbronn and Cassels about the…

Applying Littlewood's lemma in connection to Riemann's Hypothesis and exploiting the symmetry of Riemann's $xi$ function we show that almost all nontrivial Riemann's Zeta zeros are on the critical line.

In this manuscript, we consider the Riemann zeta function $\zeta$, defined through the Abel summation formula. We present a simple analytical method based on a complex differential equation. The aim is to propose a new analytical approach,…

Some problems involving the classical Hardy function $$ Z(t) := \zeta(1/2+it)\bigl(\chi(1/2+it)\bigr)^{-1/2}, \quad \zeta(s) = \chi(s)\zeta(1-s) $$ are discussed. In particular we discuss the odd moments of $Z(t)$, the distribution of its…

For $0 < a \le 1/2$, we define the quadrilateral zeta function $Q(s,a)$ using the Hurwitz and periodic zeta functions and show that $Q(s,a)$ satisfies Riemann's functional equation studied by Hamburger, Heck and Knopp. Moreover, we prove…