Erdős Problem #420

declared status 'open'. Formalized: no. If $\tau(n)$ counts the number of divisors of $n$ then let$$F(f,n)=\frac{\tau((n+\lfloor f(n)\rfloor)!)}{\tau(n!)}.$$Is it true that$$\lim_{n\to \infty}F((\log n)^C,n)=\infty$$for large $C$? Is it true that $F(\log n,n)$ is everywhere dense in $(1,\infty)$? More generally, if $f(n)\leq \log n$ is a monotonic function such that $f(n)\to \infty$ as $n\to \infty$, then is $F(f,n)$ everywhere dense? Current best: Erd\H{o}s, Graham, Ivi\'{c}, and Pomerance [EGIP96] have proved that$$\liminf F(c\log n, n) = 1$$for any $c>0$, and$$\lim F(n^{4/9},n)=\infty.$$(The exponent $4/9$ can be improved slightly.) They also prove that, if $f(n)=o((\log n)^2)$, then for almost all $n$$$F(f,n)\sim 1.$$van Doorn notes in the comments that the existence of infinitely many bounded prime gaps implies$$\limsup_{n\to \infty}F(g(n),n)=\infty$$for any $g(n)\to \infty$, and that Cram\'{e}r's conjecture implies$$\lim F(g(n)(\log n)^2, n)=\infty$$for any $g(n)\to \infty$> References [EGIP96] Erd\H{o}s, Paul and Graham, S. Prize: no. Tags: number theory.

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