Functional equations for double series of Euler type with coefficients

Abstract

We first survey the known results on functional equations for the double zeta-function of Euler type and its various generalizations. Then we prove two new functional equations for double series of Euler-Hurwitz-Barnes type with complex coefficients. The first one is of general nature, while the second one is valid when the coefficients are Fourier coefficients of a cusp form.

Environmental Justice and Circular Economy: Analyzing Justice for Waste Pickers in Upcoming Circular Economy in Fortaleza, Brazil

“…Lastly, a tentative prognosis is offered in the conclusion. Both Walker's problemframing structure and Schlosberg's EJ theory [32] set up the backbone of this paper; Schlosberg identifies three key issues to be considered-distribution, participation, and recognition-briefly explained below.…”

“…At the time the author conducted the interviews, there were nine registered WP associations, and all were contacted, as was the head of their collective network REDE. All ten were interviewed, with the exception of one-due to [32,58] and Walker [31] Circular Economy and Sustainability schedule incompatibility. Representatives of both the state and municipal governments were also interviewed.…”

A Survey on the Theory of Multiple Dirichlet Series with Arithmetical Coefficients as Numerators

“…can be continued. This double series satisfies a certain "functional equation", which is written in terms of confluent hypergeometric functions (see [10,11]):…”

Double Dirichlet series associated with arithmetic functions

“…We further calculate reverse values of Φ 2 (s 1 , s 2 ; 1, Λ) at points on the sets of singularity (see Propositions 5.1 and 5.4 and Example 5.3), and also those of Φ 2 (s 1 , s 2 ; 1, µ) (see Example 6.1). If α is an arithmetic function for which Φ(s; α) has only finitely many poles, then the double series of the form (1.8) has been studied by Choie and the first-named author [4]. However Φ(s; α) defined by (1.7) obviously has infinitely many poles, so is outside of the study in [4].…”

“…If α is an arithmetic function for which Φ(s; α) has only finitely many poles, then the double series of the form (1.8) has been studied by Choie and the first-named author [4]. However Φ(s; α) defined by (1.7) obviously has infinitely many poles, so is outside of the study in [4].…”