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

[jean-paul allouche]

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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