From Basel problem to multiple zeta values
type: Item
instance of: scholarly article
title: From Basel problem to multiple zeta values
author: Masato Kobayashi
published in: The Journal of Analysis
publication date: 2024-12-05T00:00:00Z
review text: Let \(\zeta\) denote the Riemann zeta function.The aim of the paper under review is to show the following integral evaluations\N\begin{align*}\N\int_0^1 \frac{\arcsin x \arccos x}{x} \, dx &= \frac{7}{8} \zeta(3), \\\N\frac{2}{\pi} \int_0^1 \frac{\arcsin^2 x \arccos x}{x} \, dx &= \frac{1}{8} \zeta(3),\N\end{align*}\Nand discuss their applications to central binomial series, multiple zeta values, and some other sums shown as Theorems 2.5, 2.6, 2.7, 2.11, 2.16, 2.23. The main idea is to evaluate integrals involving powers of arcsine function.\N\NTheorem 2.6.\NLet \(G\) denote the Catalan constant\N\[\NG = 1 - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + \cdots \approx 0.915966 \dots\N\]\NThen\N\begin{align*}\N\int_0^1 \frac{\mathrm{arcsinh} \, x}{\sqrt{1-x^2}} \, dx &= G, \\\N\int_0^1 \frac{\mathrm{arcsinh} \, x\arccos x}{x} \, dx &= \frac{\pi^3}{32}.\N\end{align*}\N\NTheorem 2.7.\N\[\N\int_0^1 \frac{\arctan x \operatorname{arcctg} x}{x} \, dx = \frac{7}{8} \zeta(3).\N\]\N\NFor positive integers \(i_1, \dots, i_k\) such that \(i_1 \geq 2\), define the multiple zeta value and multiple \(t\)-value by\N\begin{align*}\N\zeta(i_1, i_2, \dots, i_k) &= \sum_{n_1 > n_2 > \cdots > n_k > 0} \frac{1}{n_1^{i_1} n_2^{i_2} \cdots n_k^{i_k}}, \\\Nt(i_1, i_2, \dots, i_k) &= \sum_{\substack{n_1 > n_2 > \cdots > n_k > 0 \\ n_j \text{ odd}}} \frac{1}{n_1^{i_1} n_2^{i_2} \cdots n_k^{i_k}}.\N\end{align*}\NLet \(\{m\}^n\) denote the sequence \((m, m, \dots, m)\) of length \(n\).\N\NTheorem 2.16.\NFor \(n \geq 0\),\N\begin{align*}\N\zeta(3, \{2\}^n) &= \frac{2}{\pi} \int_0^1 \frac{2^{2n+3}\arcsin^{2n+2} x \arccos x}{ (2n+2)! x} \, dx, \\\Nt(3, \{2\}^n) &= \int_0^1 \frac{\arcsin^{2n+1} x \arccos x}{(2n+1)! x} \, dx.\N\end{align*}\N\NThe author finishes the article with the following conjecture.\N\NConjecture 2.26.\N\[\N\sum_{k=0}^\infty \frac{\binom{2k}{k}}{2^{2k} (2k+1)^4} = \frac{1}{48} \left( 6\pi \zeta(3) + 4\pi \log^3 2 + \pi^3 \log 2 \right).\N\]\N\NRecently, quite similar series w…
reviewed by: Valeriĭ V. Volchkov
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 7954776
zbMATH Keywords: Apéry constant
zbMATH Keywords: arcsine function
zbMATH Keywords: Basel problem
zbMATH Keywords: Maclaurin series
zbMATH Keywords: multiple zeta values
zbMATH Keywords: Riemann zeta function
zbMATH Keywords: Wallis integral
MaRDI profile type: Publication
cites work: Q4184941
cites work: Q3218192
cites work: An Elementary Proof of ∑ ∞ n = 1 1/n 2 = π 2 /6
cites work: Integer powers of arcsin
cites work: On the Evaluation of Euler Sums
cites work: An odd variant of multiple zeta values
cites work: Q3351551
cites work: Interesting Series Involving the Central Binomial Coefficient
cites work: Another look at Zagier's formula for multiple zeta values involving Hoffman elements
cites work: On Hoffman's \(t\)-values of maximal height and generators of multiple zeta values
cites work: (Almost) impossible integrals, sums, and series. With a foreword by Paul J. Nahin
cites work: Evaluation of the multiple zeta values \(\zeta(2,\ldots,2,3,2,\ldots,2)\)
DOI: 10.1007/S41478-023-00682-4
description: scientific article; zbMATH DE number 7954776