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Re: Additive-Subtractive Generation of Primes
Posted:
Mar 9, 2009 2:57 PM
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>From Osher Doctorow
In the first paragraph of the last post, the first occurrence of the words "successive primes p(n) or something like that should have referred to P(n), the "Prime Sum" defined later. Also see Wolfram's "Prime Zeta Function."
Osher Doctorow
---- mdoctorow@ca.rr.com wrote:
> From Osher Doctorow
>
> One of the most interesting things relating to the additive nature of prime numbers is the "duality" of successive primes p(n) (the Prime Zeta Function in which sum 1/n^p for any fixed number p in replaced by P(n) = the modified Riemann Zeta Function with sum 1/n^s replaced by sum 1/pk^n where the sum is over only primes p for all primes, with pk the kth prime also denoted p_k) with the Lucas numbers Ln:
>
> 1) ln(0.3739558136...) = - [sum(Ln - 1)P(n)]/n, sum from n = 2 to infinity
>
> where quantity on the left whose logarithm is being tkaen is Artin's Constant C_ARTIN. It is true that C_ARTIN is obtained ordinarily from an infinite product equation involving pn and pn - 1 where pn is the nth prime, but since it is a constant, (1) really does express a type of "duality" between P(n) and Ln or Ln - 1, when averaged by n.
>
> P(s) can also be defined as:
>
> 2) P(s) = sum 1/p^s, sum over all primes p
>
> See Wolfram's (Eric Weisstein's) "Prime Sums" for more on
> ---- mdoctorow@ca.rr.com wrote:
> > From Osher Doctorow
> >
> > So what are primes "for", that is to say what they they "do" as analogs of "atoms" in physics, in mathematics if they are not basically multiplicative divisive?
> >
> > In other words, intuitively primes have always seemed to come from factoring integers or fractions, as for example:
> >
> > 1) 50 = 2 x 5^2 (uniquely up to order of factors)
> >
> > What other main role could they have in mathematics and physics?
> >
> > Here is arguably a clue. Look at the multiplicative inverses of primes:
> >
> > 2) 1/2, 1/3, 1/5, 1/7, 1/11, 1/13, ...
> >
>
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