Special values of the Riemann zeta function via arcsine random variables

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...

Some Unusual Identities for Special Values of the Riemann Zeta Function ∗

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

Values of the Riemann zeta function at integers

Ladder Heights , Gaussian Random Walks , and the Riemann Zeta Function

Yale University and University of California, Berkeley Let Sn n ≥ 0 be a random walk having normally distributed increments with mean θ and variance 1, and let τ be the time at which the random walk first takes a positive value, so that Sτ is the first ladder height. Then the expected value EθSτ, originally defined for positive θ, may be extended to be an analytic function of the complex variab...

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...