# Asymptotic behaviour of the partition function

## Abstract

Given a pair of positive integers m and d such that 2{<=}m{<=}d, for integer n{>=}0 the quantity b{sub m,d}(n), called the partition function is considered; this by definition is equal to the cardinality of the set. The properties of b{sub m,d}(n) and its asymptotic behaviour as n{yields}{infinity} are studied. A geometric approach to this problem is put forward. It is shown that C{sub 1}n{sup {lambda}{sub 1}}{<=}b{sub m,d}(n){<=}C{sub 2}n{sup {lambda}{sub 2}}, for sufficiently large n, where C{sub 1} and C{sub 2} are positive constants depending on m and d. For some pair (m,d) the exponents {lambda}{sub 1} and {lambda}{sub 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{sub 1} and C{sub 2} are obtained.

- Authors:

- M.V. Lomonosov Moscow State University, Moscow (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 21202924

- Resource Type:
- Journal Article

- Journal Name:
- Sbornik. Mathematics

- Additional Journal Information:
- Journal Volume: 191; Journal Issue: 3; Other Information: DOI: 10.1070/SM2000v191n03ABEH000464; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMPTOTIC SOLUTIONS; MATHEMATICAL LOGIC; PARTITION FUNCTIONS

### Citation Formats

```
Protasov, V Yu.
```*Asymptotic behaviour of the partition function*. United States: N. p., 2000.
Web. doi:10.1070/SM2000V191N03ABEH000464; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Protasov, V Yu.
```*Asymptotic behaviour of the partition function*. United States. doi:10.1070/SM2000V191N03ABEH000464; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Protasov, V Yu. Sun .
"Asymptotic behaviour of the partition function". United States. doi:10.1070/SM2000V191N03ABEH000464; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
```

```
@article{osti_21202924,
```

title = {Asymptotic behaviour of the partition function},

author = {Protasov, V Yu},

abstractNote = {Given a pair of positive integers m and d such that 2{<=}m{<=}d, for integer n{>=}0 the quantity b{sub m,d}(n), called the partition function is considered; this by definition is equal to the cardinality of the set. The properties of b{sub m,d}(n) and its asymptotic behaviour as n{yields}{infinity} are studied. A geometric approach to this problem is put forward. It is shown that C{sub 1}n{sup {lambda}{sub 1}}{<=}b{sub m,d}(n){<=}C{sub 2}n{sup {lambda}{sub 2}}, for sufficiently large n, where C{sub 1} and C{sub 2} are positive constants depending on m and d. For some pair (m,d) the exponents {lambda}{sub 1} and {lambda}{sub 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{sub 1} and C{sub 2} are obtained.},

doi = {10.1070/SM2000V191N03ABEH000464; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)},

journal = {Sbornik. Mathematics},

issn = {1064-5616},

number = 3,

volume = 191,

place = {United States},

year = {2000},

month = {4}

}