Problem 685

Erdős Problem #685: declared status 'open'. Formalized: no. Let $\epsilon>0$ and $n$ be large depending on $\epsilon$. Is it true that for all $n^\epsilon<k\leq n^{1-\epsilon}$ the number of distinct prime divisors of $\binom{n}{k}$ is$$(1+o(1))k\sum_{k<p<n}\frac{1}{p}?$$Or perhaps even when $k \geq (\log n)^c$? Current best: It is trivial that the number of prime factors is$$>\frac{\log \binom{n}{k}}{\log n},$$and this inequality becomes (asymptotic) equality if $k>n^{1-o(1)}$. Prize: no. Tags: binomial coefficients, number theory, primes.