Asymptotic behaviour of the~partition function

V. Yu. Protasov

Sbornik. Mathematics, Tome 191 (2000) no. 3, pp. 381-414

Résumé

Given a pair of positive integers $m$ and $d$ such that $2\leqslant m\leqslant d$, for integer $n\geqslant 0$ the quantity $b_{m,d}(n)$, called the partition function is considered; this by definition is equal to the cardinality of the set $$ \biggl\{(a_0,a_1,\dots):n=\sum_ka_km^k,\ a_k\in\{0,\dots,d-1\},\ k\geqslant 0\biggr\}. $$ The properties of $b_{m,d}(n)$ and its asymptotic behaviour as $n\to\infty$ are studied. A geometric approach to this problem is put forward. It is shown that $$ C_1n^{\lambda_1}\leqslant b_{m,d}(n)\leqslant C_2n^{\lambda_2}, $$ for sufficiently large $n$, where $C_1$ and $C_2$ are positive constants depending on $m$ and $d$, and $\lambda_1=\varliminf\limits_{n\to\infty}\dfrac{\log b(n)}{\log n}$ and $\lambda_2=\varlimsup\limits_{n\to\infty}\dfrac{\log b(n)}{\log n}$ are characteristics of the exponential growth of the partition function. For some pair $(m,d)$ the exponents $\lambda_1$ and $\lambda_2$ are calculated as the logarithms of certain algebraic numbers; for other pairs the problem is reduced to finding the joint spectral radius of a suitable collection of finite-dimensional linear operators. Estimates of the growth exponents and the constants $C_1$ and $C_2$ are obtained.