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 2009-01-12, 01:48 #1 davar55     May 2004 New York City 3×17×83 Posts Find the Value It's well known that: Product[prime p > 1]{1/(1-1/p^s)} = Sum[integer n > 0]{1/n^s} = zeta(s) for real s > 1, and also that for s=2 this equals pi^2 / 6. Using this or any other way, find the value of the product for s=2 if the primes > 1 are replaced by the composites > 1. (This isn't hard.) You'll find, for s=2, the product over primes > the product over composites. Is this true for all values of s > 1, or is there some s where the two products are equal?
 2009-01-13, 22:18 #2 mart_r     Dec 2008 you know...around... 63910 Posts If my humble 5-minute-program is right, there should be a crossover point between s=1 and s=2, somewhere near sqrt(2). When s=1.395, the product over the composites gets bigger than the one with the primes. Haven't got the time right now to check s=1.4 far enough, but I guess the product over the composites will get bigger eventually as well.
 2009-01-14, 17:13 #3 mart_r     Dec 2008 you know...around... 27F16 Posts I spent another half hour or so during my lunch break today, and if my extrapolations are even roughly correct, then the point where both products are equal should be about s=1.39773 ± 0.0001. The easiest way to calculate the series of composites for me is Product (n>=2) {1/(1-1/n^s)} - Zeta(s), so I don't think I can get any further ATM (maybe another decimal digit or two at most). But it's an interesting problem nonetheless.
2009-01-14, 19:36   #4
wblipp

"William"
May 2003
New Haven

3·787 Posts

Quote:
 Originally Posted by davar55 find the value of the product for s=2 if the primes > 1 are replaced by the composites > 1. (This isn't hard.)
Hint:

The union of {composites > 1} and {primes > 1} is {integers > 1}

2009-01-14, 21:02   #5
mart_r

Dec 2008
you know...around...

63910 Posts

Quote:
 Originally Posted by wblipp Hint: The union of {composites > 1} and {primes > 1} is {integers > 1}

The answer is 12/Pi² or 1.2158542037080532573265535585...

Also, just noticed the typo in my previous post. It's not ... - Zeta(s) but ... / Zeta(s), of course!

Last fiddled with by mart_r on 2009-01-14 at 21:05

 2009-02-05, 15:16 #6 davar55     May 2004 New York City 3·17·83 Posts Nice work. I'd just like to confirm: did you determine that the cross-over point is the only one? There are no others as s --> 1 ?
2009-02-05, 17:04   #7
mart_r

Dec 2008
you know...around...

32·71 Posts

Quote:
 Originally Posted by davar55 I'd just like to confirm: did you determine that the cross-over point is the only one? There are no others as s --> 1 ?
I didn't see any other cross-over points from what I've calculated.

 2009-07-02, 19:46 #8 davar55     May 2004 New York City 3×17×83 Posts Suppose the limit is Exactly 1.4. What else (...) does that imply?

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