Irrationality of values of the Riemann zeta function

Irrationality of values of the Riemann zeta function

The paper deals with a generalization of Rivoal’s construction, which enables one to construct linear approximating forms in 1 and the values of the zeta function ζ(s) only at odd points. We prove theorems on the irrationality of the number ζ(s) for some odd integers s in a given segment of the set of positive integers. Using certain refined arithmetical estimates, we strengthen Rivoal’s origin...

Irrationality of Values of Zeta-function

1. Introduction. The irrationality of values of the zeta-function ζ(s) at odd integers s ≥ 3 is one of the most attractive problems in number theory. Inspite of a deceptive simplicity and more than two-hundred-year history of the problem, all done in this direction can easily be counted. It was only 1978, when Apéry [A] obtained the irrationality of ζ(3) by a presentation of " nice " rational a...

Values of the Riemann zeta function at integers

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...

Mollifying the Riemann Zeta-function