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

Olivier Gerard olivier.gerard at gmail.com
Tue Dec 8 04:32:45 CET 2020


Dear Gordon,

You are mistaken. Zeta(2) and similar sums of reciprocals are not about a
"natural density". Please lookup the definition.

As Jean-Paul said already, the natural density of the squares (and any
subset of them) is 0.

Olivier


On Tue, Dec 8, 2020 at 6:28 AM Gordon Speagle <gspeagle at gmail.com> wrote:

> 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
> > >
> > > --
> > > Seqfan Mailing list - http://list.seqfan.eu/
> >
> >
> > --
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> >
>
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