The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann z...

In this article we compute a discrete mean value of the derivative of the Riemann zeta function. This mean value will be important for several applications concerning the size of ζ(ρ) where ζ(s) is the Riemann zeta function and ρ is a non-trivial zero of the Riemann zeta function.

In this paper we treat the classical Riemann zeta function as a of three variables: one is usual complex 1 -dimensional, customly denoted s , another two are infinite dimensional, denote them b = { n } ∞ and z . When gets function. Our goal in to study meromorphic continuation ζ ( ) triple

We consider the modified q-analogue of Riemann zeta function which is defined by ζq(s)= ∑∞ n=1(qn(s−1)/[n]s), 0< q < 1, s ∈ C. In this paper, we give q-Bernoulli numbers which can be viewed as interpolation of the above q-analogue of Riemann zeta function at negative integers in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Also, we will treat some...

The Riemann zeta process is a stochastic process {Z(σ), σ > 1} with independent increments and marginal distributions whose characteristic functions are proportional to the Riemann zeta function along vertical lines < s = σ . We establish functional limit theorems for the zeta process and other related processes as arguments σ approach the pole at s = 1 of the zeta function (from above).

1 1 Preliminaries 1 1.1 Function fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Primes and Divisors . . . . . . . . . . . . . . . . . . . . 2 1.2.2 The Picard Group . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 Riemann-Roch . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Notation . . . . ...

In this brief note we bring out the analogy between the arithmetic of elliptic curves and the Riemann zeta-function. · · · The aim of this note is to point out the possibility of developing the theory of elliptic curves so that the arithmetical and analytic aspects are developed in strict analogy with the classical theory of the Riemann zeta-function. This has been triggered by Lemmermeyer’s pe...