Related papers: Jost function, prime numbers and Riemann zeta func…

The secondary zeta function $Z(s)=\sum_{n=1}^\infty\alpha_n^{-s}$, where $\rho_n=\frac12+i\alpha_n$ are the zeros of zeta with $\Im(\rho)>0$, extends to a meromorphic function on the hole complex plane. If we assume the Riemann hypothesis…

We discuss analogues of the prime number theorem for a hyperbolic rational map f of degree at least two on the Riemann sphere. More precisely, we provide counting estimates for the number of primitive periodic orbits of f ordered by their…

In this paper, we extend the result of Fujii on the second moment of $S(t+h) - S(t)$ to longer range of $h$ under the Riemann Hypothesis and an quantitative form of the Twin Prime Conjecture.

In the first part we present the number theoretical properties of the Riemann zeta function and formulate the Riemann Hypothesis. In the second part we review some physical problems related to this hypothesis: the links with Random Matrix…

We study the value distribution of the Riemann zeta function near the line $\Re s = 1/2$. We find an asymptotic formula for the number of $a$-values in the rectangle $ 1/2 + h_1 / (\log T)^\theta \leq \Re s \leq 1/2+ h_2 /(\log T)^\theta $,…

We investigate the statistical distribution of the zeros of Dirichlet $L$--functions both analytically and numerically. Using the Hardy--Littlewood conjecture about the distribution of prime numbers we show that the two--point correlation…

In these lectures we first review the important properties of the Riemann $\zeta$-function that are necessary to understand the nature and importance of the Riemann hypothesis (RH). In particular this first part describes the analytic…

We prove a multidimensional extension of Selberg's central limit theorem for $\log\zeta$, in which non-trivial correlations appear. In particular, this answers a question by Coram and Diaconis about the mesoscopic fluctuations of the zeros…

We propose a model-independent analysis of near-threshold enhancements using independent S-matrix poles. In this formulation, we constructed a Jost function with controllable zeros to ensure that no poles are generated on the physical…

We consider the 1D Dirac operator on the half-line with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following properties…

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 $\pi S(t)$ denote the argument of the Riemann zeta-function at the point $\frac12+it$. Assuming the Riemann Hypothesis, we sharpen the constant in the best currently known bounds for $S(t)$ and for the change of $S(t)$ in intervals. We…

By considering the prime zeta function, the author intended to demonstrate in that the Riemann zeta function zeta(s) does not vanish for Re(s)>1/2, which would have proven the Riemann hypothesis. However, he later realised that the proof of…

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…

In this paper, we discuss the value-distribution of the Riemann zeta-function. The authors give some results for the discrepancy estimate and large deviations in the limit theorem by Bohr and Jessen.

Physicists become acquainted with special functions early in their studies. Consider our perennial model, the harmonic oscillator, for which we need Hermite functions, or the Laguerre functions in quantum mechanics. Here we choose a…

The distribution of the zeros of the Euler double zeta-function $\zeta_2(s_1,s_2)$, in the case when $s_1=s_2$, is studied numerically. Some similarity to the distribution of the zeros of Hurwitz zeta-functions is observed.

We study the resonances of (generally, non-selfadjoint) Schr\"odinger operators with matrix-valued square-well potentials. We compute explicitly the Jost function and derive complex transcendental equations for the resonances. We prove…

We investigate the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence. The Hilbert-Polya operator in this interpretation is the master matrix of the large…

Recently, we proposed an exact method for direct calculation of the Jost function for central potentials (which may have Coulombic tails) and the Jost matrix for non-central short range potentials. This method works for all real or complex…