Related papers: Correlations of eigenvalues and Riemann zeros

For the Tornheim double zeta function T(s1,s2,s3) of complex variables,we obtain its functional equations,which are new.Using the calculus of r-th order derivative of zeta(s,alpha) as a function of alpha(developed in author[7])as the…

There exists an infinite series of ratios by which one can derive the Riemann zeta function $\zeta(s)$ from Catalan numbers and central binomial coefficients which appear in the terms of the series. While admittedly the derivation is not…

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…

We introduce a new method for studying murmurations, based on random matrix theory. With this method, we exhibit murmurations or similar phenomena: assuming ratios conjectures, for elliptic curves ordered by height, quadratic twists of a…

In this paper, we introduce the resonance-correlation method to study small gaps between consecutive zeros of the Riemann zeta-function. Our method is based on a synthesis of Montgomery's pair correlation approach and the Montgomery-Odlyzko…

The properties of several functions are employed to investigate the zeros of the Riemann zeta function $\zeta(a+bi)$ $(0<a<1, b\neq 0)$. If the zeros of the zeta function have not the form $\frac{1}{2}+ib$ where $i=\sqrt{-1}$, we derive a…

A generalization of a well-known relation between the Riemann zeta function $\zeta(s)$ and Bernoulli numbers $B_n$ is obtained. The formula is a new representation of the Riemann zeta function in terms of a nested series of Bernoulli…

Assuming an averaged form of Mertens' conjecture and that the ordinates of the non-trivial zeros of the Riemann zeta function are linearly independent over the rationals, we analyze the finer structure of the terms in a well-known formula…

We formulate a parametrized uniformly absolutely globally convergent series of $\zeta$(s) denoted by Z(s, x). When expressed in closed form, it is given by Z(s, x) = (s -- 1)$\zeta$(s) + 1 x Li s z z -- 1 dz, where Li s (x) is the…

A key theorem formulated in the context of functional Mellin transforms generalizes the important relationship $\exp\mathrm{tr} M=\det\exp M$. Along with the involution symmetry of the zeta function, the theorem suggests a strategy for…

We use visible point vector identities to examine polylogarithms in the neighbourhood of the Riemann zeta function zeroes. New formulas limiting to the trivial zeroes and to the critical line on the zeta function are given. Similar results…

Let $Z(t)$ be the classical Hardy function in the theory of the Riemann zeta-function. The main result in this paper is that if the Riemann hypothesis is true then for any positive integer $n$ there exists a $t_{n}>0$ such that for…

Recently Conrey, Farmer and Zirnbauer conjectured formulas for the averages over a family of ratios of products of shifted L-functions. Their L-functions Ratios Conjecture predicts both the main and lower order terms for many problems,…

If $0 < \gamma_1 \le \gamma_2 \le \gamma_3 \le \ldots$ denote ordinates of complex zeros of the Riemann zeta-function $\zeta(s)$, then several results involving the maximal order of $\gamma_{n+1}-\gamma_n$ and the sum $$ \sum_{0<\gamma_n\le…

We investigate the horizontal distribution of zeros of the derivative of the Riemann zeta function and compare this to the radial distribution of zeros of the derivative of the characteristic polynomial of a random unitary matrix. Both…

Using elementary methods we find surprising connections between the values of the Riemann Zeta Function over integers and the fractional parts of rational powers, and a connection between the Riemann Zeta Function and the Prime Zeta…

A strategy for proving Riemann hypothesis is suggested. The vanishing of the Rieman Zeta reduces to an orthogonality condition for the eigenfunctions of a non-Hermitian operator $D^+$ having the zeros of Riemann Zeta as its eigenvalues. The…

Riemann conjectured that all the zeros of the Riemann $\Xi$-function are real, which is now known as the Riemann Hypothesis (RH). In this article we introduce the study of the zeros of the truncated sums $\Xi_N(z)$ in Riemann's uniformly…

We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in…

We establish theoretical recovery guarantees of a family of Riemannian optimization algorithms for low rank matrix recovery, which is about recovering an $m\times n$ rank $r$ matrix from $p < mn$ number of linear measurements. The…