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Old 08-24-2020, 07:46 PM   #76
d2_e4
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Re: Math puzzle for those who enjoy them...

Sorry, I did x/e^x, totally missed the -1

Gonna have a think about how to solve sum 1/prime^2, seems like an interesting one.

I assume we make use of the product formula of the zeta function, take the log of both sides to give log(pi^2/6) = some sum that includes reciprocals of primes squares then do some algebra?

Last edited by d2_e4; 08-24-2020 at 07:54 PM.
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Old 08-25-2020, 10:15 AM   #77
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Re: Math puzzle for those who enjoy them...

So, this 1/prime^s thing turned up a tad more complicated than I thought. I've gone down the rabbit hole on the prime zeta function and read the Wikipedia and found a youtube video on it. I'm lost at the point of the Mobius inversion, I have no idea how that works.

Looks like there's no closed form for it anyway and it's evaluated using numerical methods? The wiki page has the values for the first few S anyway, looks like the answer to your question is around 0.45.

Incidentally, on my online travels I also stumbled across the relationship (and derivation) of Gamma(s)*Zeta(s) = your integral, which would of course mean that your integral = Gamma(2)*Zeta(2) = 1! * Zeta(2) = pi^2/6.

Last edited by d2_e4; 08-25-2020 at 10:20 AM.
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