In the shadow of Euler's greatness: adventures in the rediscovery of an intriguing sum (Q6169276)

The theme of the paper is the series \(\sum_{n=1}^\infty H_n^2/n^2\), where \(H_n= \sum_{k=1}^n 1/k\) is the \(n^{\mathrm {th}}\) harmonic number. \textit{M.~Kneser} [``A summation problem. Solution to Problem~4305''. Am. Math. Mon. 57, No.~4, 267--268 (1950)], responding to a problem proposed by H.~F. Sandham in the Monthly, was the first person to publish the value of this series, viz., \[ \sum_{n=1}^\infty \frac{H_n^2}{n^2} = \frac{17\pi^4}{360}.\tag{1} \] Nowadays, this series goes by the name of Au-Yeung series since E.~Au-Yeung in 1993, while an undergraduate at the University of Waterloo, evaluated the series numerically and, Kneser's result unbeknownst to him, conjectured (1) on the basis of the numerical evidence he had obtained. The author details the history of the Au-Yeung series, in particular the contributions of David and Jonathan Borwein and their coauthors. He goes on to present six different proofs of (1); in each of them, a crucial step is to interchange integration and summation resp.\ the order of integration for positive (Riemann integrable) functions, which follows from Tonelli's theorem. A very detailed bibliography points to a dozen more proofs of (1) that have meanwhile appeared in the literature. David and Jonathan Borwein are erroneously referred to as brothers in the text (David was Jonathan's father). A correction [\textit{S.~M. Stewart}, Math. Intell. 44, No.~3, 192 (2022; Zbl 07711054)] acknowledges this biographical mistake.