Editing Greatest common divisor (section)

== Probabilities and expected value ==
In 1972, James E. Nymann showed that {{math|''k''}} integers, chosen independently and uniformly from {{math|{{mset|1, ..., ''n''}}}}, are coprime with probability {{math|1/''ζ''(''k'')}} as {{math|''n''}} goes to infinity, where {{math|''ζ''}} refers to the [[Riemann zeta function]].<ref name="nymann">{{cite journal |first=J. E. |last=Nymann |title=On the probability that {{math|''k''}} positive integers are relatively prime |journal=[[Journal of Number Theory]] |volume=4 |issue=5 |pages=469–473 |year=1972 |doi=10.1016/0022-314X(72)90038-8 |bibcode=1972JNT.....4..469N |doi-access=free }}</ref> (See [[coprime]] for a derivation.) This result was extended in 1987 to show that the probability that {{math|''k''}} random integers have greatest common divisor {{math|''d''}} is {{math|''d''<sup>−''k''</sup>/ζ(''k'')}}.<ref name="chid">{{cite journal |first1=J. |last1=Chidambaraswamy |first2=R. |last2=Sitarmachandrarao |title=On the probability that the values of ''m'' polynomials have a given g.c.d. |journal=Journal of Number Theory |volume=26 |issue=3 |pages=237–245 |year=1987 |doi=10.1016/0022-314X(87)90081-3 |doi-access=free }}</ref>

Using this information, the [[expected value]] of the greatest common divisor function can be seen (informally) to not exist when {{math|1=''k'' = 2}}. In this case the probability that the GCD equals {{math|''d''}} is {{math|''d''<sup>−2</sup>/''ζ''(2)}}, and since {{math|1=''ζ''(2) = π<sup>2</sup>/6}} we have
: <math>\mathrm{E}( \mathrm{2} ) = \sum_{d=1}^\infty d \frac{6}{\pi^2 d^2} = \frac{6}{\pi^2} \sum_{d=1}^\infty \frac{1}{d}.</math>

This last summation is the [[Harmonic series (mathematics)|harmonic series]], which diverges. However, when {{math|''k'' ≥ 3}}, the expected value is well-defined, and by the above argument, it is
: <math> \mathrm{E}(k) = \sum_{d=1}^\infty d^{1-k} \zeta(k)^{-1} = \frac{\zeta(k-1)}{\zeta(k)}. </math>

For {{math|1=''k'' = 3}}, this is approximately equal to 1.3684. For {{math|1=''k'' = 4}}, it is approximately&nbsp;1.1106.