scholarly journals Dirichlet product and the multiple Dirichlet series over function fields

2020 ◽  
Vol 55 (2) ◽  
pp. 253-265
Author(s):  
Yoshinori Hamahata ◽  
Keyword(s):  
Dirichlet Series ◽  
Function Fields ◽  
Arithmetic Functions ◽  
Multiple Dirichlet Series ◽  
Dirichlet Product

We define the Dirichlet product for multiple arithmetic functions over function fields and consider the ring of the multiple Dirichlet series over function fields. We apply our results to absolutely convergent multiple Dirichlet series and obtain some zero-free regions for them.

scholarly journals The multiple Dirichlet product and the multiple Dirichlet series

2017 ◽  
Vol 13 (08) ◽  
pp. 2181-2193 ◽  
Author(s):  
Tomokazu Onozuka
Keyword(s):  
Dirichlet Series ◽  
Free Region ◽  
Series Expression ◽  
Multiple Dirichlet Series ◽  
Dirichlet Product

First, we define the multiple Dirichlet product and study the properties of it. From those properties, we obtain a zero-free region of a multiple Dirichlet series and a multiple Dirichlet series expression of the reciprocal of a multiple Dirichlet series.

scholarly journals TWO GENERALIZATIONS OF THE BUSCHE–RAMANUJAN IDENTITIES

2013 ◽  
Vol 09 (05) ◽  
pp. 1301-1311 ◽  
Author(s):  
LÁSZLÓ TÓTH
Keyword(s):  
Dirichlet Series ◽  
Arithmetic Functions ◽  
Symmetric Polynomials ◽  
Several Variables ◽  
Dirichlet Convolution ◽  
Multiple Dirichlet Series

We derive two new generalizations of the Busche–Ramanujan identities involving the multiple Dirichlet convolution of arithmetic functions of several variables. The proofs use formal multiple Dirichlet series and properties of symmetric polynomials of several variables.

scholarly journals Generalized multiple Dirichlet series and generalized multiple polylogarithms

2006 ◽  
Vol 124 (2) ◽  
pp. 139-158 ◽  
Author(s):  
Kohji Matsumoto ◽  
Hirofumi Tsumura
Keyword(s):  
Dirichlet Series ◽  
Multiple Polylogarithms ◽  
Multiple Dirichlet Series

Statement B and the Yang-Baxter Equation

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Statistical Mechanics ◽  
Dirichlet Series ◽  
Weyl Group ◽  
Transfer Matrix ◽  
Baxter Equation ◽  
Partition Functions ◽  
Alternate Proof ◽  
Crystal Basis ◽  
Exactly Solved Models ◽  
Multiple Dirichlet Series

This chapter reinterprets Statements A and B in a different context, and yet again directly proves that the reinterpreted Statement B implies the reinterpreted Statement A in Theorem 19.10. The p-parts of Weyl group multiple Dirichlet series, with their deformed Weyl denominators, may be expressed as partition functions of exactly solved models in statistical mechanics. The transition to ice-type models represents a subtle shift in emphasis from the crystal basis representation, and suggests the introduction of a new tool, the Yang-Baxter equation. This tool was developed to prove the commutativity of the row transfer matrix for the six-vertex and similar models. This is significant because Statement B can be formulated in terms of the commutativity of two row transfer matrices. This chapter presents an alternate proof of Statement B using the Yang-Baxter equation.

Type A Weyl Group Multiple Dirichlet Series

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Positive Integer ◽  
Dirichlet Series ◽  
Weyl Group ◽  
Algebraic Number ◽  
Type A ◽  
Algebraic Number Field ◽  
Basis Vector ◽  
Roots Of Unity ◽  
Finite Set ◽  
Multiple Dirichlet Series

This chapter describes Type A Weyl group multiple Dirichlet series. It begins by defining the basic shape of the class of Weyl group multiple Dirichlet series. To do so, the following parameters are introduced: Φ‎, a reduced root system; n, a positive integer; F, an algebraic number field containing the group μ‎₂ₙ of 2n-th roots of unity; S, a finite set of places of F containing all the archimedean places, all places ramified over a ℚ; and an r-tuple of nonzero S-integers. In the language of representation theory, the weight of the basis vector corresponding to the Gelfand-Tsetlin pattern can be read from differences of consecutive row sums in the pattern. The chapter considers in this case expressions of the weight of the pattern up to an affine linear transformation.

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