User:Michael Hardy/Petry on Euler's zeta function

Posted by David Petry on July 15th, 1995:

Subject: Euler's zeta function

Note that

: $\backslash sin\backslash left(n\backslash arcsin\backslash left(\backslash frac\; x\; n\backslash right)\backslash right)$

is a polynomial for odd n.

[ ... ]

the power series for it starts off

: $x\; -\; \backslash frac\{1\; -\; \backslash frac\{n^2\}\{6\}\}\{6\}\; x^3\; +\; \backslash cdots\; .$

[ ... ]

the roots of the polynomial are:

:

and we can factor the polynomial as

: $x\backslash prod\_i\; \backslash left(\; 1\; -\; \backslash frac\{x^2\}\{r\_i^2\}\; \backslash right)$

where r_i are the positive roots.

Equating the power series and the expansion of the product gives

: $\backslash frac\{1\; -\; \backslash frac\{1\}\{n^2\}\}\{6\}\; =\; \backslash sum\_\{m=1\}^\{(n-1)/2\}\; \backslash frac\{1\}\{\backslash left(n\backslash sin\backslash left(\; \backslash frac\{m\backslash pi\}\{n\}\; \backslash right)\backslash right)^2\; \}$

and note that

: $\backslash lim\_\{n\backslash to\backslash infty\}\; n\backslash sin\backslash left(\; \backslash frac\{m\backslash pi\}\{n\}\; \backslash right)\; =m\backslash pi$

for any fixed m.

and with just a little more work we have the desired result!