Rapidly converging series for \(\zeta(2 n + 1)\) from Fourier series
type: Item
instance of: scholarly article
title: Rapidly converging series for \(\zeta(2 n + 1)\) from Fourier series
zbMATH Open document ID: 1469.11283
DOI: 10.1155/2014/457620
author: Junesang Choi
published in: Abstract and Applied Analysis
publication date: 2019-02-14T00:00:00Z
review text: Summary: Ever since Euler first evaluated \(\zeta(2)\) and \(\zeta(2 m)\), numerous interesting solutions of the problem of evaluating the \(\zeta(2 m)\) (\(m \in \mathbb N\)) have appeared in the mathematical literature. Until now no simple formula analogous to the evaluation of \(\zeta(2 m)\) (\(m \in \mathbb N\)) is known for \(\zeta(2 m + 1)\) (\(m \in \mathbb N\)) or even for any special case such as \(\zeta(3)\). Instead, various rapidly converging series for \(\zeta(2 m + 1)\) have been developed by many authors. Here, using Fourier series, we aim mainly at presenting a recurrence formula for rapidly converging series for \(\zeta(2 m + 1)\). In addition, using Fourier series and recalling some indefinite integral formulas, we also give recurrence formulas for evaluations of \(\beta(2 m + 1)\) and \(\zeta(2 m)\) (\(m \in \mathbb N\)), which have been treated in earlier works.
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11Y60
zbMATH DE Number: 7022416
zbMATH Keywords: Fourier series
zbMATH Keywords: zeta function
Wikidata QID: Q59038378
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/457620
OpenAlex ID: W1600722537
cites work: Q3021451
cites work: Leonhard Euler: The first St. Petersburg years (1727--1741)
cites work: Q4234366
cites work: Q5834107
cites work: The Series Σ∞k = 1k-s, s = 2,3,4,⋯, Once More
cites work: Q5804312
cites work: Q4040824
cites work: Elementary Evaluation of ζ(2n)
cites work: Six Ways to Sum a Series
cites work: A New Method of Evaluating ζ(2n)
cites work: Q4021268
cites work: Q3218192
cites work: Euler and the Zeta Function
cites work: Q3916824
cites work: Application of the Hurwitz Zeta Function to the Evaluation of Certain Integrals
cites work: Q3879185
cites work: Q4184941
cites work: La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs
cites work: One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational
cites work: Some rapidly converging series for $\zeta (2n+1)$
cites work: Some families of rapidly convergent series representations for the zeta functions
cites work: Certain relationships among polygamma functions, Riemann zeta function and generalized zeta function
cites work: Q4515165
cites work: From Fourier Series to Rapidly Convergent Series for Zeta(3)
cites work: Some definite integrals associated with the Riemann zeta function
cites work: A recurrence formula for 𝜁(2𝑛)
cites work: A recursive formula for even order harmonic series
description: scientific article; zbMATH DE number 7022416