# Another bad reference, and another exercise

My name turns out to be involved in another case of problematic bibliographic reference, but in contrast with the previous case, I am completely blameless. The book Mathematical Constants turns out to contain, as Added in Press, the statement that the “probability” that two integers belonging to the ring Z[j] (where j is a primiteve root of unity of order 3) are coprime is equal to

$\frac{6}{\pi^2 H}$

where

$H=\sum_{k\geq 0}{\Bigl(\frac{1}{(3k+1)^2}-\frac{1}{(3k+2)^2}\Bigr)}$

(probability is in the usual not-quite-strictly-correct sense of looking at the limit of the proportion in larger and larger balls or boxes centered at 0). For this, a reference is given to a putative unpublished note of mine entitled “Coprimality and squarefreeness within quadratic fields”.

However, this note does not exist, and never did! The only reason for this (somewhat casual) claim is that I had explained how to prove this (it is really an exercise, in line with this one) on the NMBRTHRY mailing list back in 2003.