Abstract. We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function [11] to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested by Bogomolny and Keating [6] taking inspiration from semi-classical methods. At that point they did not write out the answer explicitly, so we do that here, illustrating that by our method all the lower order terms down to the constant can be calculated rigourously if one assumes the ratios conjecture of Conrey, Farmer and Zirnbauer. Bogomolny and Keating [4] returned to their previous results simultaneously with this current work, and have written out the full expression. The result presented in this paper agrees precisely with their formula, as well as with our numerical computations, which we include here. We also include an alternate proof of the triple correlation of eigenvalues from random U(N) matrices which follows a nearly identical method to that for the Riemann zeros, but is based on

Abstract. Results of extensive computations of moments of the Riemann zeta function on the critical line are presented. Calculated values are compared with predictions motivated by random matrix theory. The results can help in deciding between those and competing predictions. It is shown that for high moments and at large heights, the variability of moment values over adjacent intervals is substantial, even when those intervals are long, as long as a block containing 109 zeros near zero number 1023. More than anything else, the variability illustrates the limits of what one can learn about the zeta function from numerical evidence. It is shown the rate of decline of extreme values of the moments is modelled relatively well by power laws. Also, some long range correlations in the values of the second moment, as well as asymptotic oscillations in the values of the shifted fourth moment, are found. The computations described here relied on several representations of the zeta function. The numerical comparison of their effectiveness that is presented is of independent interest, for future large scale computations. 1.

We give a natural extension of the classical definition of Césaro convergence of a divergent sequence/function. This involves understanding the spectrum of eigenvalues and eigenvectors of a certain Césaro operator on a suitable space of functions or sequences. The essential idea is applicable in identical fashion to other summation methods such as Borel’s. As an example we show how to obtain the analytic continuation of the Riemann zeta function ζ(z) for Re z ≤ 1 directly from generalised Césaro summation of its divergent defining series. We discuss a variety of analytic and symmetry properties of these generalised methods and some possible further applications. 1.

Riemann hypothesis, universality & broken symmetry

For the classical hamiltonian (x +1/x)(p +1/p), with position x and conjugate momentum p,allorbitsarebounded. Afterasymmetrization,thecorresponding quantum integral equation possesses a family of self-adjoint extensions: compact operators on the entire positive x axis, labelled by an angle α specifying the boundary condition at the origin, with a discrete spectrum of real energies E.Onthecylinder{− ∞ < E < ∞,0 � α<2π},thereisasingleeigencurvein the form of a helix winding clockwise. The rise between successive windings gets sharper as the scaled Planck’s constant decreases. This behaviour can be understood semiclassically. The first two terms of the asymptotic eigenvalue density are the same as those for the density of heights of the Riemann zeros. PACS numbers: 02.30.Mv, 02.30.Tb, 03.65.Ge, 03.65.Sq (Some figures in this article are in colour only in the electronic version)

Hilbert and Pólya put forward the idea that the zeros of the Riemann zeta function may have a spectral origin: the values of tn such that 1 2 + itn is a non trivial zero of ζ might be the eigenvalues of a self-adjoint operator. This would imply the Riemann Hypothesis. From the perspective of Physics one might go further and consider the possibility that the operator in question corresponds to the quantization of a classical dynamical system. The first significant evidence in support of this spectral interpretation of the Riemann zeros emerged in the 1950’s in the form of the resemblance between the Selberg trace formula, which relates the eigenvalues of the Laplacian and the closed geodesics of a Riemann surface, and the Weil explicit formula in number theory, which relates the Riemann zeros to the primes. More generally, the Weil explicit formula resembles very closely a general class of Trace Formulae, written down by Gutzwiller, that relate quantum energy levels to classical periodic orbits in chaotic Hamiltonian systems. The second

Number variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric α-stable processes.

In this paper we propose a quantum gauge system from which we construct generalized Wilson loops which will be as quantum knots. From quantum knots we give a classification table of knots where knots are one-to-one assigned with an integer such that prime knots are bijectively assigned with prime numbers and the prime number 2 corresponds to the trefoil knot. Then by considering the quantum knots as periodic orbits of the quantum system and by the identity of knots with integers and an approach which is similar to the quantum chaos approach of Berry and Keating we derive a trace formula which may be called the von Mangoldt-Selberg-Gutzwiller trace formula. From this trace formula we then give a proof of the Riemann Hypothesis. For our proof of the Riemann Hypothesis we show that the Hilbert-Polya conjecture holds that there is a self-adjoint operator for the nontrivial zeros of the Riemann zeta function and this operator is the Virasoro energy operator with central charge c = 1 2. Our approach for proving the Riemann Hypothesis can also be extended to prove the Extended Riemann Hypothesis. We also investigate the relation of our approach for proving the Riemann Hypothesis with the Random Matrix Theory for L-functions.

Abstract not found

The Riemann zeta function ζ(s) is defined by ζ(s) = ∑∞ n=1 1 ns for ℜ(s)> 1 and can be extended to a regular function on the whole complex plane deleting its unique pole at s = 1. The Riemann hypothesis is a conjecture made by Riemann in 1859 asserting that all non-trivial zeros for ζ(s) lie on the line ℜ(s) = 1 2, which is equivalent to the prime number theorem in the form of π(x)−Li(x) = O(x 1 2 +ǫ) for any positive ǫ, where π(x) = ∑ p≤x 1 with the sum runs through the set of primes is the prime counting function and Li(x) = ∫ x 1 2 log v dv is Gauss ’ logarithmic integral function. In this article, it gives a proof for the density hypothesis and so that settles the long time due justification for the Riemann hypothesis from the equivalence of the density hypothesis and the Riemann hypothesis proved recently in [12], which in turn gives a prime number theorem stated as above.