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Topic:
using cumulative eulerproduct as an approximation for finite harmonic series
Replies:
4
Last Post:
Jan 20, 2012 7:28 AM




Re: using cumulative eulerproduct as an approximation for finite harmonic series
Posted:
Jan 6, 2012 10:15 AM


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} (1p^{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 EulerMascharoni 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 eulerconstant 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



