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\title{Some Mathematics for Essay 2}
\date{\today}
\author{Fred Thulin}
\begin{document}
  \maketitle
  
  Here are some values of the $\zeta$ function and an integral that we have seen
are related to Essays 1 or 2. The numerical approximations were computed by Wolfram Alpha
(e.g. \url{http://www.wolframalpha.com/input/?i=\zeta(3/2)}).
The volume of the Ultra-Deluxe Set, $\displaystyle \zeta\left(\frac32\right)$,
``is employed in calculating the critical temperature for a Bose-Einstein condensate
in a box with periodic boundary conditions, and for spin wave physics in magnetic systems,''
according to Wikipedia [1]. The volume of the Deluxe Set, $\displaystyle \zeta(3)$,
is known as Ap\'ery's constant after the mathematician that first proved it to be irrational.
See Wikipedia [1] for the above and further information, and also Sondow [2].
  \begin{align*}
  \zeta\left(\frac32\right)&=\sum_{n=1}^\infty \frac{1}{n^\frac32}\approx
  2.6123753486854883433485675679240716305708
  \\
  \zeta(2)&=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}6\approx
1.6449340668482264364724151666460251892189499
  \\
  \zeta(3)&=\sum_{n=1}^\infty \frac{1}{n^3}\approx
  1.20205690315959428539973816151144999
  \\
  V_\text{horn}&=\pi\int_1^\infty \frac1{x^\frac32}dx=
  \left.-\frac{2\pi}{\sqrt{x}}\right|_1^\infty=2\pi\approx
6.283185307179586476925286766559
  \end{align*}
The horn volume is for the $\displaystyle\frac1{x^\frac34}$ horn of revolution [3].

Note that the approximation given for $\zeta(\frac32)$ is not a sufficient argument
that the UDS volume is less than 3. One proper way to get this result is to show that
\[
\sum_{n=2}^\infty \frac{1}{n^\frac32}<\int_1^\infty \frac1{x^\frac32}dx
\text{, and so}\sum_{n=1}^\infty \frac{1}{n^\frac32}<1+\int_1^\infty \frac1{x^\frac32}dx
\text{.}
\]
Evaluating the integral (=2) thus gives the required inequality for the UDS volume.

\newpage
{\Large References}\\
\rule{5in}{.01in}

\begin{enumerate}
\item Riemann zeta function. (2010, October 3).\\
In {\em Wikipedia, The Free Encyclopedia}.\\
Retrieved 20:17, October 15, 2010, from\\
\url{http://en.wikipedia.org/w/index.php?title=Riemann_zeta_function\&oldid=388491076}
\item Sondow, Jonathan and Weisstein, Eric W. Riemann Zeta Function.\\
In {\em MathWorld -- A Wolfram Web Resource}.\\
Retrieved October 15, 2010, from\\
\url{http://mathworld.wolfram.com/RiemannZetaFunction.html}
\item Thulin, Fred. {\em Some Mathematics for Essay 2.}\\
Retrieved October 15, 2010, from\\
\url{http://math.uic.edu/~fthulin/essay2math.pdf}
\end{enumerate}

\end{document}
​