This paper describes basic properties of the Riemann zeta function and its generalizations, indicates some of geometric analogies, and presents various formulations of the Riemann hypothesis. It briefly discusses the approach of A. Connes to a “spectral” interpretation of the Riemann zeros via noncommuative geometry, which is treated in detail by Paula Tretkoff [33].

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 paper we give new estimates for integrals involving some arithmetic functions defined over prime numbers. The main focus here is on the counting function $\pi(x)$ and Chebyshev $\vartheta$-function. Some of these depend correctness Riemann hypothesis nontrivial zeros zeta $\zeta(s)$.

Under the Riemann hypothesis, we use the distribution of zeros of the zeta function to get a lower bound for the maximum of some derivative of Hardy’s function.

As analogue to special values at positive integers of the Riemann zeta function, we consider Carlitz zeta values ζC(n) at positive integers n. By constructing t-motives after Papanikolas, we prove that the only algebraic relations among these characteristic p zeta values are those coming from the Euler-Carlitz relations and the Frobenius p-th power relations.

The theory of S2(δ) family of probability distributions is used to give a derivation of the functional equation of the Riemann xi function. The δ deformation of the xi function is formulated in terms of the S2(δ) distribution and shown to satisfy Riemann’s functional equation. Criteria for simplicity of roots of the xi function and for its simple roots to satisfy the Riemann hypothesis are form...

A rigorous proof is given of the hypergeometric-like representation of the Riemann zeta function discovered by K. Maslanka.

In this paper, we use elementary methods to derive some new identities for special values of the Riemann zeta function.

This paper is fundamentally a review, a thesis, of principal results obtained in some sectors of Number Theory and String Theory of various authoritative theoretical physicists and mathematicians. Precisely, we have described some mathematical results regarding the Fermat’s Last Theorem, the Mellin transform, the Riemann zeta function, the Ramanujan’s modular equations, how primes and adeles ar...