Fun fact: the sum of over all relatively prime pairs of positive integers
,
is rational. This is not true if you remove the “relatively prime” restriction.
Short proof… that sum equals
where is the Riemann zeta function. Since
is equal to
times a rational when
is positive even, the
s cancel and the result is rational, in fact
. If you remove the restriction you get
which is irrational. You can make other combinations of
s so that the
s cancel to get other rational infinite summations, for example the sum of
where
is squarefree and divides
should give
, and in both examples replacing the exponent 2 by any even integer also works.
Follow RSS/Atom feed or twitter for updates.