pbillet
Posts:
29
From:
paris
Registered:
9/23/09
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Re: Number theory kurtosis and skewness and smib
Posted:
Aug 26, 2010 1:26 AM
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> On Aug 25, 3:53 pm, cliclic...@freenet.de wrote:
> > pbillet schrieb:
> >
> >
> >
> >
> >
> > > let N(n) be the set of n first prime numbers (for
> example
> > > N(10)=(2,3,5,7,11,13,17,19,23,29)).
> >
> > > Now, we can consider skewness and kurtosis of N.
> >
> > > In resultku, I computed (with smib) kurtosis of N
> for integers from
> > > 2 to 50002 (with a step of 500 integers); in
> resultsk, same with
> > > skewness (It took one hour...).
> >
> > > For kurtosis, it seems to tend to -1,215... "as n
> tends to
> > > infinity", the sequence seems to oscillate around
> this value
> >
> > > For skewness, the sequence is decreasing "as n
> tends to infinity",
> > > but I haven't idea of limit.
> >
> > > Are there results on this subject?
> >
> > There may be no closed expression, but I expect
> that you will get more
> > accurate results much faster if you base your
> integration on an
> > analytical expression for the asymptotic
> prime-number density. I think
> > these analytical density functions were
> 19th-century findings by Gauß
> > (conjecture), Riemann (proof), and probably others.
> >
> > Martin.
>
> Can use prime(n) ~ n*log(n) and get what might be a
> reasonable
> approximation. Below is in Mathematica, where
> Kurtosis definition is 3
> larger than that used in the original post.
>
> In[307]:= Timing[Kurtosis[Table[Prime[n], {n, 1,
> 1000000}] // N] - 3]
> Out[307]= {2.16267, -1.21303}
>
> In[308]:= Timing[Kurtosis[Table[Log[n]*n, {n, 1.,
> 1000000.}]] - 3]
> Out[308]= {0.146977, -1.21341}
>
> Daniel Lichtblau
> Wolfram Research
Thank you for your answers, but you didn't really answer my question (maybe not well posed). In fact, here moments of first and second order are infinite, but those limits are finite.
What is the interpretation ?
Best regards,
Philippe Billet.
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