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Asymptotic behaviour of the partition function
V. Yu. Protasov M. V. Lomonosov Moscow State University
Abstract:
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.
UDC:
511
MSC: Primary
11P81; Secondary
47A13 Received: 23.06.1999
DOI:
10.4213/sm464
Fulltext:
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