A triple integral analog of a multiple zeta value
| Autor: | Victor H. Moll, Christophe Vignat, Armin Straub, Tewodros Amdeberhan |
|---|---|
| Přispěvatelé: | Department of Mathematics [Tulane, New Orleans], Tulane University |
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Integral representation Mathematics - Number Theory 010308 nuclear & particles physics Multiple integral Mathematics::Number Theory 010102 general mathematics 01 natural sciences Riemann zeta function [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] symbols.namesake Mathematics - Classical Analysis and ODEs 0103 physical sciences symbols Classical Analysis and ODEs (math.CA) FOS: Mathematics Number Theory (math.NT) 0101 mathematics Value (mathematics) ComputingMilieux_MISCELLANEOUS MathematicsofComputing_DISCRETEMATHEMATICS Mathematics |
| Zdroj: | International Journal of Number Theory International Journal of Number Theory, World Scientific Publishing, 2021, 17 (02), pp.223-237. ⟨10.1142/S1793042120400102⟩ |
| ISSN: | 1793-0421 |
| DOI: | 10.1142/S1793042120400102⟩ |
| Popis: | We establish the triple integral evaluation \[ \int_{1}^{\infty} \int_{0}^{1} \int_{0}^{1} \frac{dz \, dy \, dx}{x(x+y)(x+y+z)} = \frac{5}{24} \zeta(3), \] as well as the equivalent polylogarithmic double sum \[ \sum_{k=1}^{\infty} \sum_{j=k}^{\infty} \frac{(-1)^{k-1}}{k^{2}} \, \frac{1}{j \, 2^{j}} = \frac{13}{24} \zeta(3). \] This double sum is related to, but less approachable than, similar sums studied by Ramanujan. It is also reminiscent of Euler's formula $\zeta(2,1) = \zeta(3)$, which is the simplest instance of duality of multiple polylogarithms. We review this duality and apply it to derive a companion identity. We also discuss approaches based on computer algebra. All of our approaches ultimately require the introduction of polylogarithms and nontrivial relations between them. It remains an open challenge to relate the triple integral or the double sum to $\zeta(3)$ directly. Comment: 15 pages |
| Databáze: | OpenAIRE |
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