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Citations and links

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Citations and links

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1 Sums of reciprocals

Euler found the sum of the reciprocals of the squares in 1734 [3]. Theorem 1 states it; a modern account is in [1], and the story of the problem in [2]. The proof in Section 2 rests on the product formula (1).

Theorem 1 (Euler [3]). P 𝑛 ≥ 1 1 𝑛 2 = 𝜋 2 6 .

2 The proof

The sine is the product

(1) sin 𝑥 𝑥 = Q 𝑛 ≥ 1 ( 1 − 𝑥 2 𝑛 2 𝜋 2 ) .

Comparing the coefficients of 𝑥 2 on both sides gives the theorem; the last step is written out below. Why the problem has the name it has is in the last section.

The coefficient of 𝑥 2 is − 1 / 6 on the left of (1) and − P 𝑛 ≥ 1 1 / ( 𝑛 2 𝜋 2 ) on the right.

3 The name

Basel was the home of the Bernoullis, who made the problem known [2]. More on the Basel problem at https://en.wikipedia.org; displayed formulas in general on the page Displays.

References

[1] M. Aigner and G. M. Ziegler. Proofs from THE BOOK, 6th edition. Springer, 2018.

[2] W. Dunham. Euler: The Master of Us All. MAA, 1999.

[3] L. Euler. De summis serierum reciprocarum. Commentarii academiae scientiarum Petropolitanae 7 (1740), 123–134.

\section{Sums of reciprocals}\label{sec:sums}
Euler found the sum of the reciprocals of the squares in
1734~\cite{euler1740}. \autoref{thm:basel} states it; a modern account is
in~\cite{aigner2018}, and the story of the problem in~\cite{dunham1999}.
The proof in Section~\ref{sec:proof} rests on the product
formula~\eqref{eq:product}.

\begin{theorem}[Euler~\cite{euler1740}]\label{thm:basel}
$\displaystyle \sum_{n \ge 1} \frac{1}{n^2} = \frac{\pi^2}{6}$.
\end{theorem}

\section{The proof}\label{sec:proof}
The sine is the product
\begin{equation}\label{eq:product}
  \frac{\sin x}{x} = \prod_{n \ge 1} \Bigl(1 - \frac{x^2}{n^2 \pi^2}\Bigr).
\end{equation}
Comparing the coefficients of $x^2$ on both sides gives the theorem; the
\hyperlink{last-step}{last step} is written out below. Why the problem has
the name it has is in \hyperref[sec:name]{the last section}.

\hypertarget{last-step}{The coefficient} of $x^2$ is $-1/6$ on the left
of~\eqref{eq:product} and $-\sum_{n \ge 1} 1/(n^2 \pi^2)$ on the right.

\section{The name}\label{sec:name}
Basel was the home of the Bernoullis, who made the problem
known~\cite{dunham1999}. More on
\href{https://en.wikipedia.org/wiki/Basel_problem}{the Basel problem} at
\url{https://en.wikipedia.org}; displayed formulas in general on the page
\href{../displays/}{Displays}.

\begin{thebibliography}{9}
\bibitem{aigner2018} M.~Aigner and G.~M.~Ziegler.
  \emph{Proofs from THE BOOK}, 6th edition. Springer, 2018.
\bibitem{dunham1999} W.~Dunham.
  \emph{Euler: The Master of Us All}. MAA, 1999.
\bibitem{euler1740} L.~Euler. De summis serierum reciprocarum.
  \emph{Commentarii academiae scientiarum Petropolitanae} 7 (1740), 123--134.
\end{thebibliography}

Point at a link to see where it goes; press it to jump. A label on another page of the same site resolves through the link map the Hugo integration writes. A reference to a label that is not published anywhere stays plain text, and so does a link to a page number, since a page without pages has nowhere to send it. An address with no scheme, like ../displays/, is relative to the page.