Riemann Hypothesis and Random Walks: The Zeta Case

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...

On The Riemann Hypothesis For The Characteristic p Zeta Function

Other Representations of the Riemann Zeta Function and an Additional Reformulation of the Riemann Hypothesis

New expansions for some functions related to the Zeta function in terms of the Pochhammer's polynomials are given (coe cients bk, dk, d̂k and ˆ̂ dk). In some formal limit our expansion bk obtained via the alternating series gives the regularized expansion of Maslanka for the Zeta function. The real and the imaginary part of the function on the critical line is obtained with a good accuracy up to ...

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...