Related papers: On a Certain Integral Over a Triangle

We present several sequences of Euler sums involving odd harmonic numbers. The calculational technique is based on proper two-valued integer functions, which allow to compute these sequences explicitly in terms of zeta values only.

We evaluate binomial series with harmonic number coefficients, providing recursion relations, integral representations, and several examples. The results are of interest to analytic number theory, the analysis of algorithms, and…

Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb N$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2$…

In this paper we demonstrate the importance of a mathematical constant which is the value of several interesting numerical series involving harmonic numbers, zeta values, and logarithms. We also evaluate in closed form a number of numerical…

The set of multiple zeta-star values is a countable dense subset of the half line $(1,+\infty)$. In this paper, we establish some classical Diophantine type results for the set of multiple zeta-star values. Firstly, we give a criterion to…

Sums of squares of $|\zeta(1/2+it)|$ over short intervals are investigated. Known upper bounds for the fourth and twelfth moment of $|\zeta(1/2+it)|$ are derived. A discussion concerning other possibilities for the estimation of higher…

A relationship between the Riemann zeta function and a density on integer sets is explored. Several properties of the examined density are derived.

We estimate the maximal number of integral points which can be on a convex arc in the plane with given length, minimal radius of curvature and initial slope.

In this paper, by using the theory of circulant matrices we study some matrices over finite fields which involve the quadratic character and trinomial coefficients.

We investigate a special class of quadratic Hamiltonians on so(4) and so(3,1) and describe Hamiltonians that have additional polynomial integrals. One of the main results is a new integrable case with an integral of sixth degree.

In this paper we give criteria about estimation of derivatives of the Riemann Zeta Function on the line $\sigma=1$.

We study a family of mixed Tate motives over $\mathbb{Z}$ whose periods are linear forms in the zeta values $\zeta(n)$. They naturally include the Beukers-Rhin-Viola integrals for $\zeta(2)$ and the Ball-Rivoal linear forms in odd zeta…

We construct a converging geometric iterated function system on the moduli space of ordered triangles, for which the involved functions have geometric meanings and contain a non-contraction map under the natural metric.

We consider a geometric percolation process partially motivated by recent work of Hejda and Kala. Specifically, we start with an initial set $X \subseteq \mathbb{Z}^2$, and then iteratively check whether there exists a triangle $T \subseteq…

Hirose, Murahara, and Saito proved that some $t$-adic symmetric multiple zeta values, for indices in which $1$ and $3$ appear alternately in succession, can be expressed as polynomials in Riemann zeta values, and conjectured similar…

Multiple zeta values have been studied by a wide variety of methods. In this article we summarize some of the results about them that can be obtained by an algebraic approach. This involves "coding" the multiple zeta values by monomials in…

To an ideal in $\mathbb{C}[x,y]$ one can associate a topological zeta function. This is an extension of the topological zeta function associated to one polynomial. But in this case we use a principalization of the ideal instead of an…

We show how to obtain infinitely many continued fractions for certain Z-linear combinations of zeta and L values. The methods are completely elementary.

A {\em $1-$vertex triangulation} of an oriented compact surface $S$ of genus $g$ is an embedded graph $T\subset S$ with a unique vertex such that all connected components of $S\setminus T$ are triangles (adjacent to exactly 3 edges of $T$).…

In this article we prove some identities which allow us to evaluate some multiple unit square integrals. In our examples we will give the value of some double and triple integrals. Then, we prove several classical integral formulas with the…