Search

Versión postprint In this paper we develop an effective procedure for expressing Stark units in real quadratic extensions of totally real fields as values of the Barnes multiple Gamma function at algebraic points. This procedure is used to explicitly generate non-abelian extensions of ℚ by special Gamma values. As a main component of our work, we develop an algorithm to compute Shintani sets in all dimensions. UCR::Vicerrectoría de Docencia::Ciencias Básicas::Facultad de Ciencias::Escuela de Matemática UCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias Básicas::Centro de Investigaciones en Matemáticas Puras y Aplicadas (CIMPA)

We introduce an "$L$-function" $\mathcal{L}$ built up from the integral representation of the Barnes' multiple zeta function $\zeta$. Unlike the latter, $\mathcal{L}$ is defined on a domain equipped with a non-trivial action of a group $G$. Although these two functions differ from each other, we can use $\mathcal{L}$ to study $\zeta$. In fact, the transformation formula for $\mathcal{L}$ under $G$-transformations provides us with a new perspective on the special values of both $\zeta$ and its $s$-derivative. In particular, we obtain Kronecker limit formulas for $\zeta$ when restricted to points fixed by elements of $G$. As an illustration of this principle, we evaluate certain generalized Lambert series at roots of unity, establishing pertinent algebraicity results. Also, we express the Barnes' multiple gamma function at roots of unity as a certain infinite product. It should be mentioned that this work also considers twisted versions of $\zeta$., Comment: 27 pages

In this work, multiple gamma functions of order n have been considered. The logarithmic derivative of the multiple gamma function is known as the multiple psi function. Subadditive, superadditive, and convexity properties of higher-order derivatives of the multiple psi function are derived. Some related inequalities for these functions and their ratios are also obtained.

We consider partial zeta functions $\zeta(s,c)$ associated with ray classes $c$'s of a totally real field. Stark's conjecture implies that an appropriate product of $\exp(\zeta'(0,c))$'s is an algebraic number which is called a Stark unit. Shintani gave an explicit formula for $\exp(\zeta'(0,c))$ in terms of Barnes' multiple gamma function. Yoshida ``decomposed'' Shintani's formula: he defined the symbol $X(c,\iota)$ satisfying that $\exp(\zeta'(0,c))=\prod_{\iota} \exp(X(c,\iota))$ where $\iota$ runs over all real embeddings of $F$. Hence we can decompose a Stark unit into a product of $[F:\mathbb Q]$ terms. The main result is to show that $([F:\mathbb Q]-1)$ of them are algebraic numbers. We also study a relation between Yoshida's conjecture on CM-periods and Stark's conjecture., Comment: 30 pages

Let Γ 3 be the triple gamma function. Our main objective is to prove that the function log ⁡ Γ 3 ( z + 1 ) − Γ 3 ′ ( 1 ) z z 3 Log z + Γ 3 ′ ( 1 ) z − 1 , z ∈ C ∖ ( − ∞ , 0 ] is a Pick function and find its Stieltjes representation.

We provide new representations for the finite parts at the poles and the derivative at zero of the Barnes zeta function in any dimension in the general case. These representations are in the forms of series and limits. We also give an integral representation for the finite parts at the poles. Similar results are derived for an associated function, which we term homogeneous Barnes zeta function. Our expressions immediately yield analogous representations for the logarithm of the Barnes Gamma function, including the particular case also known as multiple Gamma function., Comment: This is the Author's Original Manuscript of an article published in Integral Transforms and Special Functions (Taylor & Francis). Published article available at http://www.tandfonline.com/eprint/ME3GywdpTKcevj9jUE5t/full (There are no differences in content between the versions.)

The theory of Barnes beta probability distributions is advanced and related to the Riemann xi function. The scaling invariance, multiplication formula, and Shintani factorization of Barnes multiple gamma functions are reviewed using the approach of Ruijsenaars and shown to imply novel properties of Barnes beta distributions. The applications are given to the meromorphic extension of the Selberg integral as a function of its dimension and the scaling invariance of the underlying probability distribution. This probability distribution in the critical case is described and conjectured to be the distribution of the derivative martingale. The Jacobi triple product is interpreted probabilistically resulting in an approximation of the Riemann xi function by the Mellin transform of the logarithm of a limit of Barnes beta distributions., 36 pages. Sections 5 and 7 of this article supersede Th. 4.5 and Section 5 of arXiv:1305.4422v1, respectively. To appear in Forum Mathematicum

The multiple gamma function Γn(z), defined by a recurrence-functional equation as a generalization of the Euler gamma function, is used in many applications of pure and applied mathematics, and theoretical physics. The theory of the multiple gamma function has been related to certain spectral functions in mathematical physics, to the study of functional determinants of Laplacians of the n-sphere, to Hecke L-functions, to the Selberg zeta function, and to the random matrix theory. There is a wide class of definite integrals and infinite sums appearing in statistical physics (the Potts model) and the lattice theory, which can be computed by means of the Γn(z) function. This paper presents new integral representations for the multiple gamma function and other mathematical functions and constants.

A new family of Barnes beta distributions on $(0, \infty)$ is introduced and its infinite divisibility, moment determinacy, scaling, and factorization properties are established. The Morris integral probability distribution is constructed from Barnes beta distributions of types $(1,0)$ and $(2,2),$ and its moment determinacy and involution invariance properties are established. For application, the maximum distributions of the 2D gaussian free field on the unit interval and circle with a non-random logarithmic potential are conjecturally related to the critical Selberg and Morris integral probability distributions, respectively, and expressed in terms of sums of Barnes beta distributions of types $(1,0)$ and $(2,2).$, 27 pages. To appear in J. Stat. Phys

Gauss' and Euler's type infinite product representations for Vignéras multiple gamma function are presented. As an application of the representations, a multiplication formula for the function is derived., 弘前大学教育学部紀要. 108, 2012, p.31-39

Selberg and Morris integral probability distributions are long conjectured to be distributions of the total mass of the Bacry-Muzy Gaussian Multiplicative Chaos measures with non-random logarithmic potentials on the unit interval and circle, respectively. The construction and properties of these distributions are reviewed from three perspectives: analytic based on several representations of the Mellin transform, asymptotic based on low intermittency expansions, and probabilistic based on the theory of Barnes beta probability distributions. In particular, positive and negative integer moments, infinite factorizations and involution invariance of the Mellin transform, analytic and probabilistic proofs of infinite divisibility of the logarithm, factorizations into products of Barnes beta distributions, and Stieltjes moment problems of these distributions are presented in detail. Applications are given in the form of conjectured mod-Gaussian limit theorems, laws of derivative martingales, distribution of extrema of $1/f$ noises, and calculations of inverse participation ratios in the Fyodorov-Bouchaud model., 64 pages

We study the asymptotic behavior of multiple Hurwitz zeta functions and generalized multiple gamma and sine functions as some period parameters tend to 0. We obtain the perfect asymptotic expansions and get some applications in refining a previous paper by the author.

We investigate values of the multiple sine function of period (1,...,1) at rational numbers. We show explicit formulas using special values of Dirichlet L-functions. We also study the case of the multiple gamma function.

Deformations of the multiple gamma and sine functions with respect to their periods are studied. To describe such deformations explicitly, a new class of generalized gamma and sine functions are introduced. In particular, we study the deformations from the viewpoint of multiplication formulas and Raabe's integral formulas for these gamma and sine functions. This new class of gamma functions contains Milnor's type multiple gamma functions as a special case.

We investigate values of the multiple sine function of period $ (1, ldots ,1) $ at rational numbers. We show explicit formulas using special values of Dirichlet $ L $-functions. We also study the case of the multiple gamma function

The second author defined the absolute CM-period symbol using the multiple gamma function. This symbol is conjectured to give the same value as Shimura's period symbol up to multiplication by algebraic numbers. In this paper, we define a p-adic analogue of the absolute CM-period symbol, using the p-adic multiple gamma function studied by the first author. In the completely split case, we present a conjecture which predicts the exact value of this symbol as well as supporting evidences.

We study continuous period deformations of the multiple Hurwitz zeta functions and their derivatives. Moreover we investigate period deformations of the generalized multiple gamma and sine functions and give applications.

We study gamma and sine functions attached to a finite dimensional linear representation of a Lie group including their multiple versions via zeta regularizations. These definitions allow us to understand several multiple gamma and sine functions known so far, such as Barnes' gamma functions, Shintani's sine functions, and their q-analogues, in a unified way. Further, we discuss period integrals for such gamma functions.

The multiple gamma function Γ n , defined by a recurrence-functional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma function has been revived. This paper discusses some theoretical aspects of the Γ n function and their applications to summation of series and infinite products.

The differential algebraicity of the multiple sine functions \(S_{r} (x, \underline{\omega})\) is investigated. The goal of the paper is to show the differential algebraicity of \(S_{r} (x, \underline{\omega})\) when the period \(\underline{\omega})\) is “rational” by using the classical theory due to Ostrowski.

We show that Shintani's work on multiple zeta and gamma functions can be simplified and extended by exploiting difference equations. We re-prove many of Shintani's formulas and prove several new ones. Among the latter is a generalization to the Shintani–Barnes gamma functions of Raabe's 1843 formula ∫ 0 1 log Γ(x) dx= log 2π , and a further generalization to the Shintani zeta functions. These explicit formulas can be interpreted as “vanishing period integral” side conditions for the ladder of difference equations obeyed by the multiple gamma and zeta functions. We also relate Barnes’ triple gamma function to the elliptic gamma function appearing in connection with certain integrable systems.

We define a q-analogue of the basic multiple gamma function [Formula: see text] introduced in [17] which differs from the one defined by Barnes [1] via the zeta regularized product. We call it a q-basic multiple gamma function. Using this q-basic multiple Gamma function we introduce a q-analogue of the multiple sine function of order m + 1. We study properties of such functions from the periodicity, differential-difference equations and duplication formulas, etc. points of view.

The authors give several new (and potentially useful) relationships between the multiple Gamma functions and other mathematical functions and constants. As by-products of some of these relationships, a classical definite integral due to Euler and other definite integrals are also considered together with closed-form evaluations of some series involving the Riemann and Hurwitz Zeta functions.