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I am interested in the history of the Basel problem. More specifically, I'm interested in knowing the history of failed attempts before Euler's crack of it, so as to know what bits of evidence informed his approach.

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On Euler's work and rigor issues see Is it true that Euler did not prove the sum of the Basel series? Here I will focus on prior work. After Mengoli proposed the Basel problem in Novae quadraturae Arithmeticae (1650), it proceeded along two lines, numerical approximations and algebraic manipulations of similar simpler series. Infinite products also played a role, but their relevance became clear only after Euler.

Numerical approximations were non-trivial. The series converges very slowly and plainly summing its first 1000 terms would only give two correct decimal places. Nonetheless, already Wallis got three correctly in Arithmetica infinitorum (1656), de Moivre got nine in 1730, and Euler himself, without knowing that, got eight in 1731. The idea was to transform the series to speed up convergence, e.g. Euler's transformation was $\sum\frac1{n^2}=\sum\frac1{2^nn^2}+(\ln 2)^2$, which converges much faster. It was based on integrating the power series for $\frac{\ln(1-x)}{x}$, see Ayoub for details.

This trick followed the idea that Leibniz communicated in a 1673 letter to Johann Bernoulli, where he integrated the same series and applied integration by parts to find that summing $\sum\frac{(-1)^{n+1}}{n^2}$ reduces to computing integrals of the form $\int x^e(1+x)^n$. As he added, "if perhaps it were possible to consider all the cases in order, some light would be shed upon the problem."

The only rigorous result on the matter was also related to Leibniz. Huygens challenged him in 1672 to sum the reciprocals of the triangular numbers $\sum\frac2{n(n+1)}$. Leibniz found the sum to be $2$ by representing it as a telescoping series $\sum\left(\frac2{n}-\frac2{n+1}\right)$. In his 1689 dissertation, which made the problem famous and gave it the name (it was defended in Basel), Jacob Bernoulli noticed that $\frac1{n^2}<\frac2{n(n+1)}$, and hence the Basel sum is less than $2$.

The two closest inspirations for Euler's breakthrough were Wallis's product formula for $\pi$ (which he did not relate to the Basel series despite considering both in the same book), and Leibniz's summation of the alternating harmonic series $\sum\frac{(-1)^{n+1}}{2n+1}=\frac\pi4$, based on the power series for the arctangent (derived earlier by Nilakantha and Gregory but unknown to Leibniz, see Roy, The Discovery of the Series Formula for π). They suggested the use of power series and infinite products, and a role for $\pi$.

But Euler's key idea that led to the 1734 breakthrough was highly original and without precursors. It boldly extended factorization of polynomials to power series, specifically the power series for $\frac{\sin x}x$. Euler assumed that its real roots $\pm\,\pi n$ were the only ones, and wrote $$\frac{\sin x}x=\prod\left(1-\frac{x^2}{\pi^2 n^2}\right),$$ as if it was a polynomial. Formally distributing in the infinite (!) product, gave the coefficient of $x^2$ on the right as the sum of the Basel series times $-\frac1{\pi^2}$. The coefficient on the left was known to be $-\frac16$ from the power series for $\sin x$, which gives the Basel sum as $\frac{\pi^2}6$. "If only my brother were still alive" vowed Johann Bernoulli (referring to Jacob, who passed away in 1705) when Euler communicated the result to him.

Euler was aware that his brilliant trick was shaky, so he applied it to $1-\sin x$, and... got the $\frac\pi4$ for the Leibniz's series from 1674. He also applied the above product formula for $\sin x$ to get Wallis's product for $\pi$. That so boosted his confidence that what he remarked next sounds like taken straight from a book of fallacies:

"For our method, which may appear to some as not reliable enough, a great confirmation comes here to light. Therefore, we should not doubt at all of the other things which are derived by the same method."

And yet he doubted still, giving two more proofs, in 1745 and 1755.

References:

Some details are given in Ayoub, Euler and the Zeta Function and Calinger, Leonhard Euler: The First St. Petersburg Years (1727–1741). Dunham's When Euler Met l’Hopital describes influences on Euler's third (1755) derivation of the Basel sum. Dunham's book Euler, The Master of Us All (the title quoting Gauss) is also a great source on context and details. Siu, Euler and Heuristic Reasoning and The Use of Many Independent Lines of Evidence: The Basel Problem post on Less Wrong describe the more empirical science approach to mathematics that prevailed before and during Euler's time and influenced his thinking and discoveries.

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    $\begingroup$ I would also imagine that, at the time, the excellent (well, perfect) numerical agreement was taken as powerful evidence that the conclusion itself was correct (even if there were gaps in the reasoning). :) $\endgroup$ Aug 15, 2022 at 23:30

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