On a question of Douglass and Ono

It is known that the partition function $p(n)$ obeys Benford's law in any integer base $b\ge 2$. In a recent paper, Douglass and Ono asked for an explicit version of this result. In my talk, I will show that for any string of digits of length $f$ in base $b$, there is $n\le N(b,f)$, where 

$N(b,f):=\exp\left(10^{32} (f+11)^2(\log b)^3\right)$

such that $p(n)$ starts with the given string of digits in base $b$. The proof uses a lower bound for a nonzero linear form in logarithms of algebraic numbers with algebraic coefficients due to  Philippon and Waldschmidt. A similar result holds for the plane partition function.