On the Digits of Partition Functions
Siddharth Iyer
TL;DR
The paper studies the problem of determining the least $n$ such that the partition function $p(n)$ or plane partitions $PL(n)$ begins with a given digit string $f$ in base $b$. It develops a digit-prefix framework using maps $C_{b,t}$ and fractional-part conditions on the logarithms, leveraging precise asymptotics and error estimates to certify leading digits. The main contributions are explicit, improved bounds: $N_p(f,b) leq 290 b^{2t} / log^2 b$ and $N_{PL}(f,b) leq 29396 b^{3t/2} / log^{3/2} b$, which surpass Luca's previous results. This advances understanding of base-$b$ leading-digit behavior for partition-type sequences and provides a practical method to bound the first occurrence of a given digit prefix.
Abstract
We address a problem of Douglas and Ono concerning the determination of an upper bound for the smallest integer $n$ such that the partition function of $n$ begins with a string $f$ of digits in base $b$. Here we improve previous results of Luca.