Sharp bounds for the partition function of integer sequences
type: Item
instance of: scholarly article
title: Sharp bounds for the partition function of integer sequences
zbMATH Open document ID: 0628.10014
DOI: 10.1007/BF01937358
author: Daniel C. Mayer
published in: BIT
publication date: 1987-01-01T00:00:00Z
review text: Let \(f=\{f_ 1,f_ 2,...\}\) be an infinite strictly increasing sequence of positive integers and define the partition function p of the sequence f in the following way. For any integer n, let p(n) denote the number of solutions of the equation \(n=f_{i_ 1}+...+f_{i_ s}\) with varying \(s\geq 0\) and indices \(i_ 1<...<i_ s\). \textit{K. F. Roth} and \textit{G. Szekeres} [Q. J. Math., Oxf. II. Ser. 5, 241-259 (1954; Zbl 0057.039)] presented a sufficient condition for the validity of an asymptotic expansion for the partition function p of the sequence f. As to the behaviour of p for small arguments, several authors obtained results by the application of a theorem of \textit{H.-E. Richert} [Norsk Mat. Tidsskr. 31, 120-122 (1949; Zbl 0040.308)]. The support of p may contain infinitely many integers and even all the integers greater than some minimal value. In the last case, the numbers \(b(m)=\inf \{b\geq 0|\min_{n\geq b}p(n)\geq m\}\) for \(m\geq 1\) are sharp lower bounds for the partition function p of f. The criteria of \textit{J. W. S. Cassels} [Acta Sci. Math. 21, 111-124 (1960)], \textit{P. Erdős} [Acta Arith. 7, 345-354 (1962; Zbl 0106.038)] and \textit{J. Folkman} [Can. J. Math. 18, 643-655 (1966; Zbl 0151.037)] give sufficient conditions for b(1) to be finite. Let C(m) denote the following condition. C(m): For suitable integers \(k\geq 0\), \(a\geq -1\) and \(d\geq f_{k+1}\), every integer n of \([a+1,a+d]\) has at least m representations by sums of distinct terms of \(\{f_ 1,...,f_ k\}\). H.-E. Richert [ibid.] proved that condition C(1) and the growth-condition \(f_{i+1}\leq 2f_ i\) for all \(i\geq k+1\) yield the estimate \(b(1)\leq a+1\). Theorem 1 of the paper under review asserts that condition C(m) and the growth-condition \(f_{k+i+1}\leq d+f_{k+1}+...+f_{k+i}\) for all \(i\geq 1\) yield the estimate \(b(m)\leq a+1\). Theorem 1 implies Richert's theorem (Corollary 1) and \textit{W. Sierpiński}'s theorem [Ann. Mat. Pura Appl., IV. Ser. 39, 69-74 (1955; Zb…
Mathematics Subject Classification ID: 11P81
Mathematics Subject Classification ID: 11P05
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 4021228
zbMATH Keywords: partition function of sequence of positive integers
zbMATH Keywords: higher multiplicities of representations
zbMATH Keywords: examples
zbMATH Keywords: tables
reviewed by: Mihály Szalay
MaRDI profile type: Publication
cites work: Q5780772
cites work: Über Zerlegungen in ungleiche Quadratzahlen
cites work: Über Zerlegungen in \(n\)-te Potenzen mit lauter verschiedenen Grundzahlen
cites work: Über Zerfällungen in ungleiche Primzahlen
cites work: Q5800402
cites work: SOME ASYMPTOTIC FORMULAE IN THE THEORY OF PARTITIONS
cites work: Sur une propriété des nombres naturels
cites work: Q3269237
cites work: Q5621399
cites work: Q3287326
cites work: Approximate formulas for some functions of prime numbers
cites work: On the representation of large integers as sums of distinct summands taken from a fixed set
cites work: Q5733621
cites work: On the Representation of Integers as Sums of Distinct Terms from a Fixed Sequence
cites work: A Stronger Bertrand's Postulate with an Application to Partitions
cites work: Addendum to "A Stronger Bertrand's Postulate with an Application to Partitions"
cites work: Q5680212
cites work: Sums of Distinct Primes from Congruence Classes Modulo 12
cites work: Primes with a Prime Subscript
cites work: Q4094915
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description: scientific article; zbMATH DE number 4021228