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Old 2009-01-12, 01:48   #1
davar55
 
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Default 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?
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Old 2009-01-13, 22:18   #2
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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.
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Old 2009-01-14, 17:13   #3
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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.
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Old 2009-01-14, 19:36   #4
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Quote:
Originally Posted by davar55 View Post
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}
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Old 2009-01-14, 21:02   #5
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Quote:
Originally Posted by wblipp View Post
Hint:


The union of {composites > 1} and {primes > 1} is {integers > 1}
Ah, completely forgot about this bit.

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
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Old 2009-02-05, 15:16   #6
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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 ?
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Old 2009-02-05, 17:04   #7
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Quote:
Originally Posted by davar55 View Post
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.
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Old 2009-07-02, 19:46   #8
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Suppose the limit is Exactly 1.4.

What else (...) does that imply?
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