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I added Euler’s second proof. I regret that Deacon Vorbis did not reply to my last message on the talk page. I have been waiting for several months for an answer. |
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Using the [[Weierstrass factorization theorem]], it can also be shown that the left-hand side is the product of linear factors given by its roots, just as we do for finite polynomials (which Euler assumed as a [[heuristic]] for expanding an infinite degree [[polynomial]] in terms of its roots, but is in general not always true for general <math>P(x)</math>):<ref>A priori, since the left-hand-side is a [[polynomial]] (of infinite degree) we can write it as a product of its roots as
:<math>\begin{align}
\sin(x) & = x (x^2-\pi^2)(x^2-4\pi^2)(x^2-9\pi^2) \cdots \\
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By Euler's proof for <math>\zeta(2)</math> explained above and the extension of his method by elementary symmetric polynomials in the previous subsection, we can conclude that <math>\zeta(2k)</math> is ''always'' a [[rational]] multiple of <math>\pi^{2k}</math>. Thus compared to the relatively unknown, or at least unexplored up to this point, properties of the odd-indexed [[zeta constants]], including [[Apéry's constant]] <math>\zeta(3)</math>, we can conclude much more about this class of [[zeta constants]]. In particular, since <math>\pi</math> and integer powers of it are [[Transcendental number|transcendental]], we can conclude at this point that <math>\zeta(2k)</math> is [[irrational]], and more precisely, [[Transcendental number|transcendental]] for all <math>k \geq 1</math>.
===Euler’s second proof===
In 1741, Euler published a second proof that did not rely on infinite products<ref>{{Cite web|url=https://faculty.math.illinois.edu/~reznick/sandifer.pdf|title=Euler’s Solution of the Basel Problem – The Longer Story|last=|first=|date=|website=https://faculty.math.illinois.edu/|url-status=live|archive-url=|archive-date=|access-date=}}</ref>. In it, he computes the integral <math> \int ^{1}_{0}\frac{\arcsin( x)}{\sqrt{1-x^{2}}} \ dx </math> by two methods: first directly, and then by expanding the arcsine as its [[Taylor series]] and integrating term-by-term. He then computes <math>\int ^{1}_{0}\frac{x^{2n+1}}{\sqrt{1-x^{2}}} \ dx</math> with <math>\displaystyle n\ \in \mathbb{N}</math> using an [[integration by parts]]. Equating the two gives the value of <math>\sum ^{+\infty }_{n=0}\frac{1}{( 2n+1)^{2}}</math>. He finally decomposes each positive integer into the product of an odd number and a power of two, then uses a [[geometric series]] to complete the proof.
Euler's proof also wasn't fully rigorous, because of issues with the interchange of the integrals and sum. The [[Monotone_convergence_theorem]] can justify this interchange.
==The Riemann zeta function ==
The [[Riemann zeta function]] {{math|''ζ''(''s'')}} is one of the most significant functions in mathematics because of its relationship to the distribution of the [[prime number]]s. The zeta function is defined for any [[complex number]] {{math|''s''}} with real part greater than 1 by the following formula:
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