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

[Allan Wechsler]

Gordon, perhaps you have misunderstood what it means to be a "square-free
integer". It's not a matter of taking all the integers and discarding the
squares. Rather, an integer is said to be "square-free" if it is not
divisible by any square greater than 1.

On Mon, Dec 7, 2020 at 1:15 PM 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 1/(2k-1)^2?
> 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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