In 1918 G. Hardy and J. Littlewood proved an asymptotic estimate for the Second moment of the modulus of the Riemann zeta-function on the segment [1/2,1/2+iT] in the complex plane, as T tends to infinity. In 1926 Ingham proved an asymptotic estimate for the fourth moment. However, since Ingham’s result, nobody has proved an asymptotic formula for any higher moment. Recently J. Conrey and A. Gho...

For an operator of a certain class in Hilbert space, we introduce axioms of an abstract intersection theory, which we prove to be equivalent to the Riemann Hypothesis concerning the spectrum of that operator. In particular if the nontrivial zeros of the Riemann zeta-function arise from an operator of this class, the original Riemann Hypothesis is equivalent to the existence of an abstract inter...

The connection between random matrix theory and the Riemann zeta function was established in 1973 when Montgomery, who had conjectured the 2-point correlations of the Riemann zeros, and Dyson, who was interested in similar statistics of the eigenvalues of ensembles of unitary matrices, realized that the formulae they had discovered independently were in fact identical in a natural asymptotic li...

Abstract. In 1997 the author [11] found a criterion for the Riemann hypothesis for the Riemann zeta function, involving the nonnegativity of certain coefficients associated with the Riemann zeta function. In 1999 Bombieri and Lagarias [2] obtained an arithmetic formula for these coefficients using the “explicit formula” of prime number theory. In this paper, the author obtains an arithmetic for...

In this note we define Riemann integrabillty for real valued functions defined on a compact metric space accompanied by a finite Borel measure. If the measure of each open ball equals the measure of its corresponding closed ball, then a bounded function is Riemann integrable if and only if its set of points of discontinuity has measure zero. Let denote the algebra of sets generated by the open ...

The Riemann zeta function is a meromorphic functionon the whole complex plane. It has infinitely many zeros and aunique pole at s = 1. Those zeros at s = −2,−4,−6, . . . areknown as trivial zeros. The Riemann hypothesis, conjectured byBernhard Riemann in 1859, claims that all non-trivial zeros of ζ(s)lie on the line R(s) =12 . The density hypothesis is a conjecturede...

The convexity properties of the kernel O. Also, lower bounds for the Tunin differences involving the moments of <ll(t) are established. The paper concludes with several questions and open problems.

Several identities for the Riemann zeta-function ζ(s) are proved. For example, if s = σ + it and σ > 0, then ∞ −∞ (1 − 2 1−s)ζ(s) s 2 dt = π σ (1 − 2 1−2σ)ζ(2σ). Let as usual ζ(s) = ∞ n=1 n −s (ℜe s > 1) denote the Riemann zeta-function. The motivation for this note is the quest to evaluate explicitly integrals of |ζ(1 2 + it)| 2k , k ∈ N, weighted by suitable functions. In particular, the prob...

It is proved that the Riemann zeta function does not satisfy any nontrivial algebraic difference equation whose coefficients are meromor-phic functions φ with Nevanlinna characteristic satisfying T (r, φ) = o(r) as r → ∞.

The series is convergent when s is a complex number with <(s) > 1. Some special values of ζ(s) are well known, for example the values ζ(2) = π/6, ζ(4) = π/90, were obtained by Euler. In 1859, Riemann had the idea to define ζ(s) for all complex number s by analytic continuation. This continuation is very important in number theory and plays a central role in the study of the distribution of prim...