[seqfan] Re: Natural Density of Odd Squares ( Euler)

[Gordon Speagle]

Hi Jean Paul,

Yes,  I meant the series 1/(2k-1)^2 converges to pi^2/8 Thanks.

 Didn't Euler show that the natural density of non square integers is
(6/pi^2) as a result of his solution to the Basel Problem?

The series of (1/n^2) converges to pi^2/6.  It's reciprocal gives the
density of square free integers.

The natural density of primes approaches zero, but the density of square
free integers and square free odd integers can be approximated with the
above identities involving pi, I think.

Thanks,

Gordon






On Mon, Dec 7, 2020, 12:15 jean-paul allouche <jean-paul.allouche at imj-prg.fr>
wrote:

> Hi
>
> I am not sure of what you mean by "1/( 2k-1)": is there a misprint?
> I guess you might have meant ?
> Anyway the density of squares is zero -- a fortiori the density of
> odd or even squares is 0 as well
>
> best wishes
> jean-paul
>
>
> Le 07/12/2020 à 17:10, Gordon Speagle a écrit :
> > Hello Everyone,
> >
> > This is my inaugural message to the group.  My name is Gordon
> >
> > I am not an academic mathematician, but in my investigations into
> numbers I
> > will sometime encounter something  that is interesting to me and I find
> > myself unsure of its role in the larger context of professional
> > mathematics.
> >
> > After revisiting Euler's solution to the Basel problem and his linking of
> > pi to the distribution of square free integers, using his same logic, it
> > can be shown that the series of 1/( 2k-1) converges to  (pi2)/ 8.
> >
> > Heuristically, it seems that this would mean that the naturally density
> of
> > non odd square integers is 8/pi2.
> >
> > Would the density of the set of the remaining odd square  integers be
> >
> > 1- 8/pi2 ?
> >
> > It seems that follows from an Erdos theorem on natural density.
> >
> > I am unsure if my reasoning is sound and would like to discuss among
> those
> > with similar interests. Which is what precipitated my post.
> >
> > Thanks
> >
> > Gordon
> >
> > --
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>
>
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