The exponential-type generating function of the Riemann zeta-function revisited

Mollifying the Riemann Zeta-function

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...

The Theory of the Riemann Zeta-function

Moments of the Riemann Zeta-function

0 |ζ( 1 2 + it)| dt. For positive real numbers k, it is believed that Mk(T ) ∼ CkT (logT ) 2 for a positive constant Ck. A precise value for Ck was conjectured by Keating and Snaith [9] based on considerations from random matrix theory. Subsequently, an alternative approach, based on multiple Dirichlet series and producing the same conjecture, was given by Diaconu, Goldfeld and Hoffstein [4]. R...

Cybernetic Interpretation of the Riemann Zeta Function

This paper uses cybernetic approach to study behavior of the Riemann zeta function. It is based on the elementary cybernetic concepts like feedback, transfer functions, time delays, PI (Proportional–Integral) controllers or FOPDT (First Order Plus Dead Time) models, respectively. An unusual dynamic interpretation of the Riemann zeta function is obtained.