Related papers: The Riemann Hypothesis for Angular Lattice Sums

We unify in a large class of additive functions the results obtained in the first part of this work. The proof rests on series involving the Riemann zeta function and certain sums of primes which may have their own interest.

By the second mean-value theorem of calculus (Gauss-Bonnet theorem) we prove that the class of functions ${\mit \Xi}(z)$ with an integral representation of the form $\int_{0}^{+\infty}du\,{\mit \Omega}(u)\,{\rm ch}(uz)$ with a real-valued…

This paper proposes a reformulation of the Riemann Xi function in order to investigate its properties. The reformulated function, which depicts the Xi function as the weighted sum of incomplete gamma functions, is validated, and a number of…

It is a well-known result that the proportion of lattice points visible from the origin is given by $\frac{1}{\zeta(2)}$, where $\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$ denotes the Riemann zeta function. Goins, Harris, Kubik and Mbirika,…

We prove that there exist infinitely many consecutive zeros of the Riemann zeta-function on the critical line whose gaps are greater than $3.18$ times the average spacing. Using a modification of our method, we also show that there are even…

Assuming the Riemann hypothesis, we obtain upper and lower bounds for moments of the Riemann zeta-function averaged over the extreme values between its zeros on the critical line. Our bounds are very nearly the same order of magnitude. The…

This research expository article contains a survey of earlier work (in \S2--\S4) but also contains a main new result (in \S5), which we first describe. Given $c \geq 0$, the spectral operator $\mathfrak{a} = \mathfrak{a}_c$ can be thought…

We show that if the derivative of the Riemann zeta function has sufficiently many zeros close to the critical line, then the zeta function has many closely spaced zeros. This gives a condition on the zeros of the derivative of the zeta…

This paper is divided into two independent parts. The first part presents new integral and series representations of the Riemaan zeta function. An equivalent formulation of the Riemann hypothesis is given and few results on this formulation…

Let $\Theta$ denote the supremum of the real parts of the zeros of the Riemann zeta function. We demonstrate that $\Theta=1$, which entails the existence of infinitely many Riemann zeros off the critical line (thus disproving the Riemann…

We study the value-distribution of the Riemann zeta-function and related functions on and near the critical line. Amongst others, we focus on the following: The critical line is a natural boundary of the Voronin-type universality property…

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,…

The function $S_n (t) = \pi \left( \frac{3}{2} - {frac} \left( \frac{\vartheta(t)}{\pi} \right) + \left( \lfloor \frac{t \ln \left( \frac{t}{2 \pi e}\right)}{2 \pi} + \frac{7}{8} \rfloor - n \right) \right)$ is conjectured to be equal to $S…

We present a quantum mechanical model which establishes the veracity of the Riemann hypothesis that the non-trivial zeros of the Riemann zeta-function lie on the critical line of $\zeta(s)$.

We show that if the Riemann Hypothesis is true, then in a region containing most of the right-half of the critical strip, the Riemann zeta-function is well approximated by short truncations of its Euler product. Conversely, if the…

The goal of this paper is to give a relatively simple proof of some known zero density estimates for Riemann zeta function which are sufficiently strong to break the density hypothesis in a nontrivial part of the critical strip. Apart from…

A detailed combinatorial analysis of planar lattice convex polygonal lines is presented. This makes it possible to answer an open question of Vershik regarding the existence of a limit shape when the number of vertices is constrained. The…

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 investigate the location of zeros and poles of a dynamical zeta function arizing in a class of lattice spin models introduced in the 60-ties by M. Kac. The transfer operator method allows us to prove the xistence of infinitely nontrivial…

A proof of the Riemann's hypothesis (RH) about the non-trivial zeros of the Riemann zeta-function is presented. It is based on the construction of an infinite family of operators D^{(k,l)} in one dimension, and their respective…