[seqfan] Natural Density of Odd Squares ( Euler)

Gordon Speagle gspeagle at gmail.com
Mon Dec 7 17:10:17 CET 2020


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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