Recall the product formula for the sine function (here and here):

Thus:

or with *z*=π*x*,

Taking the logarithm of both sides:

.

Now, , so taking the derivative of both sides of the above equation:

.

Note that via the geometric series,

.

Thus

Note that we can reverse the order of summation:

,

where is the Riemann zeta function.

Now, we use Euler’s formula: as , , , and

.

Previously, we showed that the function can be given by the series . Thus

,

and so

.

Now, as and , we can see that the first two terms of the series in the above are

.

Thus

.

Now, recall that for *n*≥2, *B*_{n}=0 for odd *n*; thus the series above only has even *n* terms, and we can rewrite it, using *n*=2*k*, as

or thus as

Comparing this to

, we see via the term-by-term comparison that

or solving for the zeta function:

which gives the Riemann zeta function for even positive integers, as previously noted here.

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Tags: Bernoulli Numbers, Geometric Series, Math, Monday Math, Riemann Zeta Function, Sine Product Formula

This entry was posted on February 9, 2009 at 12:24 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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February 11, 2009 at 9:48 am |

oh my god… I needed this several years ago

February 26, 2009 at 11:36 pm |

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