Editing Pi (section)

=== Irrationality and transcendence ===
{{See also|Proof that π is irrational{{!}}Proof that {{pi}} is irrational|Proof that π is transcendental{{!}}Proof that {{pi}} is transcendental}}
Not all mathematical advances relating to {{pi}} were aimed at increasing the accuracy of approximations. When Euler solved the [[Basel problem]] in 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between {{pi}} and the [[prime number]]s that later contributed to the development and study of the [[Riemann zeta function]]:{{sfn|Posamentier|Lehmann|2004|p=284}}

<math display=block> \frac{\pi^2}{6} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots</math>

Swiss scientist [[Johann Heinrich Lambert]] in 1768 proved that {{pi}} is [[irrational number|irrational]], meaning it is not equal to the quotient of any two integers.{{sfn|Arndt|Haenel|2006|p=5}} [[Proof that π is irrational|Lambert's proof]] exploited a continued-fraction representation of the tangent function.<ref>Lambert, Johann, "Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques", reprinted in {{harvnb|Berggren|Borwein|Borwein|1997|pp=129–140}}.</ref> French mathematician [[Adrien-Marie Legendre]] proved in 1794 that {{pi}}<sup>2</sup> is also irrational. In 1882, German mathematician [[Ferdinand von Lindemann]] proved that {{pi}} is [[transcendental number|transcendental]],<ref>{{cite journal |last=Lindemann |first=F. |author-link=Ferdinand Lindemann |year=1882 |title=Über die Ludolph'sche Zahl |journal=Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin |volume=2 |pages=679–682 |url=https://archive.org/details/sitzungsberichte1882deutsch/page/679}}</ref> confirming a conjecture made by both [[Adrien-Marie Legendre|Legendre]] and Euler.{{sfn|Arndt|Haenel|2006|p=196}}<ref>Hardy and Wright 1938 and 2000: 177 footnote §&nbsp;11.13–14 references Lindemann's proof as appearing at ''Math. Ann''. 20 (1882), 213–225.</ref> Hardy and Wright states that "the proofs were afterwards modified and simplified by Hilbert, Hurwitz, and other writers".<ref>cf Hardy and Wright 1938 and 2000:177 footnote §&nbsp;11.13–14. The proofs that e and π are transcendental can be found on pp. 170–176. They cite two sources of the proofs at Landau 1927 or Perron 1910; see the "List of Books" at pp. 417–419 for full citations.</ref>