Related papers: Iterated integrals, multiple zeta values and Selbe…
We show that integrals of the form \[ \dint_{0}^{1} x^{m}{\rm Li}_{p}(x){\rm Li}_{q}(x)dx, (m\geq -2, p,q\geq 1) \] and \[ \dint_{0}^{1} \frac{\ds \log^{r}(x){\rm Li}_{p}(x){\rm Li}_{q}(x)}{\ds x}dx, (p,q,r\geq 1) \] satisfy certain…
For a polynomial P, we consider the sequence of iterated integrals of ln P(x). This sequence is expressed in terms of the zeros of P(x). In the special case of ln(1 + x^2), arithmetic properties of certain coefficients arising are…
In this article, we study the local behaviour of the multiple zeta functions at integer points and write down a Laurent type expansion of the multiple zeta functions around these points. Such an expansion involves a convergent power series…
In this paper, we study a family of single variable integral representations for some products of $\zeta(2n+1)$, where $\zeta(z)$ is Riemann zeta function and $n$ is positive integer. Such representation involves the integral…
Starting from a result of Stewart, Tijdeman and Ruzsa on iterated difference sequences, we introduce the notion of iterated compositions of linear operations. We prove a general result on the stability of such compositions (with bounded…
We prove a kind of integral expressions for finite multiple harmonic sums and multiple zeta-star values. Moreover, we introduce a class of multiple integrals, associated with some combinatorial data (called 2-labeled posets). This class…
Since their rediscovery in the 1990s, multiple zeta values have become ubiquitous in many areas of mathematics and physics. Their standard integral and sum representations can usually be traced back to a single source, namely the iterated…
In this paper, we consider a discrete version of iterated integrals by the naive (equally divided) Riemann sum. In particular, basic three formulas for usual iterated integrals are discritized. Moreover, we proved cyclic sum formulas for…
One of the generalizations of multiple zeta values is the $q$-version, and in the case of finite sums, they may be expressed explicitly in polynomial form. Several results have been found when the powers of the factors in the denominator…
We prove several results on the distribution function of $\zeta(1+it)$ in the complex plane, that is the joint distribution function of $\arg\zeta(1+it)$ and $|\zeta(1+it)|$. Similar results are also given for $L(1,\chi)$ (as $\chi$ varies…
We define polynomials of one variable t whose values at t=0 and 1 are the multiple zeta values and the multiple zeta-star values, respectively. We give an application to the two-one conjecture of Ohno-Zudilin, and also prove the cyclic sum…
The main purpose of this paper is to present a systemic study of some families of multiple $q$-Euler numbers and polynomials. In particular, by using the $q$-Volkenborn integration on $\Bbb Z_p$, we construct $p$-adic $q$-Euler numbers and…
Certain completely logarithmic formula for a set of reversely iterated integrals (energies) is proved in this paper. Namely, in this case we have that integral powers of $\ln T$ are contained on input as well as on output of corresponding…
In this paper, we formally introduce the notion of Ap{\'e}ry-like sums and we show that every multiple zeta values can be expressed as a $\bf Z$-linear combination of them. We even describe a canonical way to do so. This allows us to put in…
In this paper, we discuss the value-distribution of the Riemann zeta-function. The authors give some results for the discrepancy estimate and large deviations in the limit theorem by Bohr and Jessen.
We introduce the method of desingularization of multi-variable multiple zeta-functions (of the generalized Euler-Zagier type), under the motivation of finding suitable rigorous meaning of the values of multiple zeta-functions at…
In this paper, we introduce the interpolated multiple $t$-values of general level and represent a generating function for sums of interpolated multiple $t$-values of general level with fixed weight, depth, and height in terms of a…
We prove a new linear relation for multiple zeta values. This is a natural generalization of the restricted sum formula proved by Eie, Liaw and Ong. We also present an analogous result for finite multiple zeta values.
A typical formula of multiple zeta values is the sum formula which expresses a Riemann zeta value as a sum of all multiple zeta values of fixed weight and depth. Recently weighted sum formulas, which are weighted analogues of the sum…
In this paper we define multiple Dedekind zeta values (MDZV), using a new type of iterated integrals, called iterated integrals on a membrane. One should consider MDZV as a number theoretic generalization of Euler's multiple zeta values.…