Wildberger's construction enables us to obtain a hypergroup from a random walk on a special graph. We will give a probability theoretic interpretation to products on the hypergroup. The hypergroup can be identified with a commutative algebra whose basis is transition matrices. We will estimate the operator norm of such a transition matrix and clarify a relationship between their matrix products and random walks.
Let 0 < a ≤ 1, s ∈ C, and ζ(s, a) be the Hurwitz zeta-function. Recently, T. Nakamura showed that ζ(σ, a) does not vanish for any 0 < σ < 1 if and only if 1/2 ≤ a ≤ 1. In this paper, we show that ζ(σ, a) has precisely one zero in the interval (0, 1) if 0 < a < 1/2. Moreover, we reveal the asymptotic behavior of this unique zero with respect to a.
We consider iterated integrals of {\log\zeta(s)} on certain vertical and horizontal lines.
Here, the function {\zeta(s)} is the Riemann zeta-function.
It is a well-known open problem whether or not the values of the Riemann zeta-function on the critical line
are dense in the complex plane.
In this paper,
we give a result for the denseness of the values of the iterated integrals on the horizontal lines.
By using this result,
we obtain the denseness of the values of {\int_{0}^{t}\log\zeta(\frac{1}{2}+it^{\prime})\,dt^{\prime}} under the Riemann Hypothesis.
Moreover, we show that, for any {m\geq 2}, the denseness of the values of an m-times iterated integral on the critical line is equivalent to the Riemann Hypothesis.
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