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Re: using cumulative eulerproduct as an approximation for finite
harmonic series
Posted:
Jan 6, 2012 10:15 AM
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On 6 jan, 14:55, "ma'ayan peres" <maayan.pe...@gmail.com> wrote:
> On Jan 6, 3:35 pm, Jean Dupont <jeandupont...@gmail.com> wrote:
>
> > I was wondering whether it is possible to use the cumulative
> > eulerproduct \prod_{p} (1-p^{-s})^{-1} replacing "for all p" with "a
> > function for a limited number of p's" to approximate \sum_{n=1}^{k}
> > \frac{1}{n} for a finite value of k
>
> > regards,
> > Jean
>
> Yes you can but perhaps it won't come out a very nice thing.
>
> What could be nicer is to realize that lim {n --> oo} (1 + 1/2 +...+ 1/
> n - log n} = gamma,
>
> with gamma = the Euler-Mascharoni constant.
>
> Tonio
yes I know, but in fact the original question was posed because of
this:
If you plot the cumulative eulerproduct function for say 10000 primes,
log x for x=1:10000 and \sum_{n=1}^{k}\frac{1}{n} for k=1 to 10000 you
see the last two graphs lying nicely the euler-constant apart but the
eulerproduct function is lying much higher, so I think the question
becomes how many primes are equivalent to how many terms in the
harmonic series?
Jean
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