Hardy–Ramanujan theorem

Hardy–Ramanujan theorem

In mathematics, the Hardy–Ramanujan theorem, proved by harvtxt|Hardy|Ramanujan|1917, states that the normal order of the number ω("n") of distinct prime factors of a number "n" is log(log("n")). Roughly speaking, this means that most numbers have about this number of distinct prime factors.

A more precise version states that for any real-valued function ψ("n") that tends to infinity as "n" tends to infinity:|omega(n)-log(log(n))|for "almost all" (all but an infinitesimal proportion of) integers. That is, let "g"("x") be the number of positive integers "n" less than "x" for which the above inequality fails: then "g"("x")/"x" converges to zero as "x" goes to infinity.)

This theorem is generalized by the Erdős–Kac theorem, which shows that in fact the distribution of distinct prime factors is essentially normal.

References

*citation|first=G.H.|last= Hardy| authorlink=G. H. Hardy| first2= S.|last2=Ramanujan|title=The normal number of prime factors of a number |journal= Quart. J. Math. |volume= 48 |year=1917|pages= 76–92
*springer|id=H/h110080|first=A.|last= Hildebrand


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