Re: The numerical value of Sum(1/(p log p))

On Mar 21, 2006 3:27 AM CT, buffalo wrote:

Thanx, Kyle. Sounds good. I didn't realize the
Dedekind Eta here. And I didn't know that Eta(1) =
ln 2. Going to have a try in my books to improve your
guess to a valid statement. :-)

Actually it's the Dirichlet eta function, I think that
the Dedekind eta is a bit different. It's actually quite
easy to show that eta(1) = ln(2), recall that

1 + x + x^2 + x^3 + ... = 1 / (1 - x) for |x| < 1.

Integrating both sides with respect to x we obtain

x + x^2 / 2 + x^3 / 3 + ... = -ln(1 - x).

Solving for the natural logarithm we find

ln(1 - x) = -x - x^2 / 2 - x^3 / 3 - ...

Plugging in -s for x it follows that

ln(1 + s) = s - s^2 / 2 + s^3 / 3 - s^4 / 4 + ....

= sum_{k = 1}^oo (-1)^(k + 1) s^k / k.

This is sometimes refered to the Mercator series. When
we started our series was valid for |x| < 1. Testing
the endpoints it can be shown that the Mercator series
converges for real s in (-1, 1]. Hence, when s = 1 we
obtain

ln(2) = sum_{k = 1}^oo (-1)^(k + 1) / k

= 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ...

...which we recognize as the Dirichlet eta function
evaluated at 1.

Regards,
Kyle Czarnecki
.

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