Growth of the product $∏^n_{j=1} (1-x^{a_j})$

• Autorzy: J. P. Bell , P. B. Borwein , L. B. Richmond
• Języki publikacji: EN

Abstrakty

EN

We estimate the maximum of $∏^n_{j=1} |1 - x^{a_j}|$ on the unit circle where 1 ≤ a₁ ≤ a₂ ≤ ... is a sequence of integers. We show that when $a_j$ is $j^k$ or when $a_j$ is a quadratic in j that takes on positive integer values, the maximum grows as exp(cn), where c is a positive constant. This complements results of Sudler and Wright that show exponential growth when $a_j$ is j.
   In contrast we show, under fairly general conditions, that the maximum is less than $2^n/n^r$, where r is an arbitrary positive number. One consequence is that the number of partitions of m with an even number of parts chosen from $a₁,...,a_n$ is asymptotically equal to the number of such partitions with an odd number of parts when $a_i$ satisfies these general conditions.

Kategorie tematyczne

• 11C08: Polynomials

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1998
• Tom: 86
• Numer: 2
• Strony: 155-170

Daty

wydano

1998

otrzymano

1997-08-19

poprawiono

1998-02-20

Twórcy

autor

J. P. Bell

• Pure Mathematics Student, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

autor

P. B. Borwein

• Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

autor

L. B. Richmond

• Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Bibliografia

• [1] F. V. Atkinson, On a problem of Erdős and Szekeres, Canad. Math. Bull. 4 (1961), 7-12.
• [2] A. S. Belov and S. V. Konyagin, On estimates for the constant term of a nonnegative trigonometric polynomial with integral coefficients, Mat. Zametki 59 (1996), 627-629 (in Russian).
• [3] P. Borwein, Some restricted partition functions, J. Number Theory 45 (1993), 228-240.
• [4] P. Borwein and C. Ingalls, The Prouhet, Tarry, Escott problem, Enseign. Math. 40 (1994), 3-27.
• [5] H. Davenport, Multiplicative Number Theory, 2nd ed., Springer, 1980.
• [6] E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34 (1979), 391-401.
• [7] P. Erdős, Problems and results on diophantine approximation, in: Asymptotic Distribution Modulo 1, J. F. Koksma and L. Kuipers (eds.), Noordhoff, 1962.
• [8] P. Erdős and G. Szekeres, On the product $∏^n_{k=1} (1-z^a_k)$, Publ. Inst. Math. (Beograd) 13 (1959), 29-34.
• [9] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford, 1979.
• [10] M. N. Kolountzakis, On nonnegative cosine polynomials with nonnegative integral coefficients, Proc. Amer. Math. Soc. 120 (1994), 157-163.
• [11] R. Maltby, Root systems and the Erdős-Szekeres Problem, Acta Arith. 81 (1997), 229-245.
• [12] A. M. Odlyzko, Minima of cosine sums and maxima of polynomials on the unit circle, J. London Math. Soc. (2) 26 (1982), 412-420.
• [13] A. M. Odlyzko and L. B. Richmond, On the unimodality of some partition polynomials, European J. Combin. 3 (1982), 69-84.
• [14] K. F. Roth and G. Szekeres, Some asymptotic formulae in the theory of partitions, Quart. J. Math. Oxford Ser. (2) 5 (1954), 241-259.
• [15] C. L. Siegel, Über die Classenzahl quadratischer Zahlkörper, Acta Arith. 1 (1935), 83-86.
• [16] C. Sudler, An estimate for a restricted partition function, Quart. J. Math. Oxford Ser. (2) 15 (1964), 1-10.
• [17] E. M. Wright, Proof of a conjecture of Sudler, Quart. J. Math. Oxford Ser., 11-15.
• [18] E. M. Wright, A closer estimation for a restricted partition function, Quart. J. Math. Oxford Ser., 283-287.