Prime zeta function statistics and Riemann zero-difference repulsion

Difference independence of the Riemann zeta function

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 → ∞.

Prime Number Theory and the Riemann Zeta-Function

Riemann ’ s zeta function and the prime number theorem

16 Riemann’s zeta function and the prime number theorem We now divert our attention from algebraic number theory to talk about zeta functions and L-functions. As we shall see, every global field has a zeta function that is intimately related to the distribution of its primes. We begin with the zeta function of the rational field Q, which we will use to prove the prime number theorem. We will ne...

A Zero Density Result for the Riemann Zeta Function

N(σ, T ) ≤ 4.9(3T ) 8 3 (1−σ) log5−2σ(T ) + 51.5 log T, for σ ≥ 0.52 and T ≥ 2000. We discuss a generalization of the method used in these two results which yields an explicit bound of a similar shape while also improving the constants. Furthermore, we present the effect of these improvements on explicit estimates for the prime counting function ψ(x). This is joint work with Habiba Kadiri and N...

q-Riemann zeta function

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...