Binomial Coefficients and the Distribution of the Primes

Abstract

Let omega(n) be the number of distinct prime factors dividing n and m > n natural numbers. We calculate a formula showing which prime numbers in which intervals divide a given binomial coefficient. From this formula we get an identity omega(binom(nk)(mk))=sum_i (pi(k/b(i))- pi(k/a(i))) + O(sqrt(k)). Erdoes mentioned that omega(binom(nk)(mk))= log n^n/(m^m (n-m)^(n-m)) k/log k + o(k/log k). As an application of the above identities, we conclude some well-known facts about the distribution of the primes and deduce for all natural numbers k an expression (also well-known) log k = sum_i a_k(i) which generalizes log 2 = sum_i^(infty) (-1)^(j+1) / j.