Monday Math 58

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

Thus:

or with zx,

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, Bn=0 for odd n; thus the series above only has even n terms, and we can rewrite it, using n=2k, 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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3 Responses to “Monday Math 58”

  1. Top entrepreneur Says:

    oh my god… I needed this several years ago

  2. Monday Math 60 « Twisted One 151’s Weblog Says:

    […] this then answers our original question: , and using and (see here), we find . Similarly, (see here), and so . Possibly related posts: (automatically generated)Euler Totient Function.Prime […]

  3. Monday Math 81 « Twisted One 151’s Weblog Says:

    […] Math 81 By twistedone151 Recall from here that . Letting and solving for the sum, we see . Now, , so the above becomes . Now, along with […]

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