Find the Value

davar55 · 2009-01-12, 01:48

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?

mart_r · 2009-01-13, 22:18

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.

mart_r · 2009-01-14, 17:13

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.

wblipp · 2009-01-14, 19:36

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}

mart_r · 2009-01-14, 21:02

Quote:

Originally Posted by wblipp

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!

davar55 · 2009-02-05, 15:16

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 ?

mart_r · 2009-02-05, 17:04

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.

davar55 · 2009-07-02, 19:46

Suppose the limit is Exactly 1.4.

What else (...) does that imply?

2009-01-12, 01:48	#1
davar55 May 2004 New York City 1000010001011₂ 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?

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2009-01-13, 22:18	#2
mart_r Dec 2008 you know...around... 2²×11×17 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... 2²·11·17 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.

2009-02-05, 15:16	#6
davar55 May 2004 New York City 108B₁₆ 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 ?

2009-07-02, 19:46	#8
davar55 May 2004 New York City 10213₈ Posts	Suppose the limit is Exactly 1.4. What else (...) does that imply?