Related papers: New exponent pairs, zero density estimates, and ze…
The alternating zeta function zeta*(s) = 1 - 2^{-s} + 3^{-s} - ... is related to the Riemann zeta function by the identity (1-2^{1-s})zeta(s) = zeta*(s). We deduce the vanishing of zeta*(s) at each nonreal zero of the factor 1-2^{1-s}…
We develop a unified density-based framework for primality, coprimality, and prime pairs, and introduce an intrinsic normalized model for prime gaps constrained by the Prime Number Theorem. Within this setting, a structural tension between…
The leading asymptotic behaviour as $t\to \infty$ of the celebrated Riemann zeta function $\zeta(s), \ s = \sigma + it, \quad 0<\sigma<1, \quad t>0 , \ t\to\infty,$ can be expressed in terms of a transcendental sum. The sharp estimation of…
We will provide an explicit log-free zero-density estimate for $\zeta(s)$ of the form $N(\sigma,T)\le AT^{B(1-\sigma)}$. In particular, this estimate becomes the sharpest known explicit zero-density estimate uniformly for…
In this paper we obtain new complete hybrid formula for corresponding class of $\zeta$-factorization formulas. We demonstrate that this formula is the synergetic one. Namely, this one describes the cooperation between some class of…
We give a number field analogue of a result of Ramanujan, Hardy and Littlewood, thereby obtaining a modular relation involving the non-trivial zeros of the Dedekind zeta function. We also provide a Riesz-type criterion for the Generalized…
A novel method is proposed to determine an analytical expression for eigenfunctions and numerical result for eigenvalues of the Schr\"odinger type equations, within the context of Taylor expansion of a function. Optimal truncation of the…
We present a method developed by the NNPDF Collaboration that allows the inclusion of new experimental data into an existing set of parton distribution functions without the need for a complete refit. A Monte Carlo ensemble of PDFs may be…
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.
A new (algebraic) approximation scheme to find {\sl global} solutions of two point boundary value problems of ordinary differential equations (ODE's) is presented. The method is applicable for both linear and nonlinear (coupled) ODE's whose…
The Bethe approximation, discovered in statistical physics, gives an efficient algorithm called belief propagation (BP) for approximating a partition function. BP empirically gives an accurate approximation for many problems, e.g.,…
With the advent of score-matching techniques for model training and Langevin dynamics for sample generation, energy-based models (EBMs) have gained renewed interest as generative models. Recent EBMs usually use neural networks to define…
This study deals with certain harmonic zeta functions, one of them occurs in the study of the multiplication property of the harmonic Hurwitz zeta function. The values at the negative even integers are found and Laurent expansions at poles…
We improve by an exponential factor the best known asymptotic upper bound for the density of sets avoiding 1 in Euclidean space. This result is obtained by a combination of an analytic bound that is an analogue of Lovasz theta number and of…
In this paper, an alternative method to range-separated linear-response time-dependent density-functional theory and perturbation theory is proposed to improve the estimation of the energies of a physical system from the energies of a…
Exact analytical expressions for available potential energy density (APE) and available potential vorticity (APV) are derived from first principles. These APE and APV expressions align with previously known quantities found using…
This paper continues a series of investigations on converging representations for the Riemann Zeta function. We generalize some identities which involve Riemann's zeta function, and moreover we give new series and integrals for the zeta…
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…
We review recent progress relating to the extreme value statistics of the characteristic polynomials of random matrices associated with the classical compact groups, and of the Riemann zeta-function and other $L$-functions, in the context…
We present a new derivation of the spectral gap of the totally asymmetric exclusion process on a half-filled ring of size L by using the Bethe Ansatz. We show that, in the large L limit, the Bethe equations reduce to a simple transcendental…