- **Puzzles**
(*https://www.mersenneforum.org/forumdisplay.php?f=18*)

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

Find the ValueIt'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? |

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. |

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. |

[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] |

[quote=wblipp;158736]Hint:
[spoiler] The union of {composites > 1} and {primes > 1} is {integers > 1} [/spoiler][/quote] Ah, completely forgot about this bit.:smile: [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! |

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 ? |

[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. |

Suppose the limit is Exactly 1.4.
What else (...) does that imply? |

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