Improved Riemann Ladder Function

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...

Riemann ’ s ζ - Function

We provide an overview of the Riemann ζ-function with an eye to a proof of the functional equation that is non-standard and more accessible than those proofs readily available in modern texts. This is s survey note with the goal of a more elegant and transparent presentation of the salient features surrounding the functional equation for the ζ-function and the intimate connections with the Riem...

Lagrangians with Riemann Zeta Function

We consider construction of some Lagrangians which contain the Riemann zeta function. The starting point in their construction is p-adic string theory. These Lagrangians describe some nonlocal and nonpolynomial scalar field models, where nonlocality is controlled by the operator valued Riemann zeta function. The main motivation for this research is intention to find an effective Lagrangian for ...

Mollifying the Riemann Zeta-function