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.
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