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-   -   Find the Value (https://www.mersenneforum.org/showthread.php?t=11320)

 davar55 2009-01-12 01:48

Find the Value

It's well known that:
Product[sub][prime p > 1][/sub]{1/(1-1/p^s)} = Sum[sub][integer n > 0][/sub]{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=davar55;158205]find the value of the product
for s=2 if the primes > 1 are replaced by the composites > 1.

(This isn't hard.)[/QUOTE]

Hint:

[spoiler]
The union of {composites > 1} and {primes > 1} is {integers > 1}
[/spoiler]

 mart_r 2009-01-14 21:02

[quote=wblipp;158736]Hint:

[spoiler]
The union of {composites > 1} and {primes > 1} is {integers > 1}
[/spoiler][/quote]

[spoiler]The answer is 12/Pi² or 1.2158542037080532573265535585...[/spoiler]

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=davar55;161661]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 ?[/quote]

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?

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