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This article is cited in 31 scientific papers (total in 31 papers)
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
$$
\{(a_0,a_1,…):n=\sum_ka_km^k, a_k\in\{0,…,d-1\}, k\geqslant 0\}.
$$
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.
DOI:
https://doi.org/10.4213/sm464
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English version:
Sbornik: Mathematics, 2000, 191:3, 381–414
Bibliographic databases:
UDC:
511
MSC: Primary 11P81; Secondary 47A13 Received: 23.06.1999
Citation:
V. Yu. Protasov, “Asymptotic behaviour of the partition function”, Mat. Sb., 191:3 (2000), 65–98; Sb. Math., 191:3 (2000), 381–414
Citation in format AMSBIB
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