On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: Denseness

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.

Integrals of logarithmic functions and alternating multiple zeta values

By using the method of iterated integral representations of series, we establish some explicit relationships between multiple zeta values and integrals of logarithmic functions. As applications of these relations, we show that multiple zeta values of the form $$\begin{array}{} \zeta ( {\bar 1,{{\left\{ 1 \right\}}_{m - 1}},\bar 1,{{\left\{ 1 \right\}}_{k - 1}}} ),\qquad (k,m\in \mathbb{N}) \end{array} $$ for m = 1 or k = 1, and $$\begin{array}{} \zeta ( {\bar 1,{{\left\{ 1 \right\}}_{m - 1}},p,{{\left\{ 1 \right\}}_{k - 1}}}),\qquad (k,m\in\mathbb{N}) \end{array} $$ for p = 1 and 2, satisfy certain recurrence relations which allow us to write them in terms of zeta values, polylogarithms and ln 2. Furthermore, we also obtain reductions for certain multiple polylogarithmic values at $\begin{array}{} \frac {1}{2} \end{array} $.

Fubini’s Theorem

Summary Fubini theorem is an essential tool for the analysis of high-dimensional space [8], [2], [3], a theorem about the multiple integral and iterated integral. The author has been working on formalizing Fubini’s theorem over the past few years [4], [6] in the Mizar system [7], [1]. As a result, Fubini’s theorem (30) was proved in complete form by this article.

New order-statistics-based ranking models and faster computation of outcome probabilities

In sport, order-statistics-based models such as Henery’s gamma model and the Thurstone–Mosteller type V model are useful in estimating competitor strengths from observed performance of players in competitions between two or more players. They can also be applied in many other areas, such as analysis of consumer preference data, which would be useful to marketing management. Two new families of such models derived from the exponentiated exponential, and Pareto distributions are introduced. Use of order-statistics-based models when there are more than two competitors has been hampered by lack of an efficient method of computation of outcome probabilities as a function of competitor strengths, and a fast method of computation of outcome probabilities is presented, which exploits the fact that the integral to be evaluated is an iterated integral.