Sums of Squares and the Preservation of Modularity under Congruence Restrictions

  • Paul T. Bateman
  • Boris A. Datskovsky
  • Marvin I. Knopp
Part of the Developments in Mathematics book series (DEVM, volume 4)

Abstract

If s is a fixed positive integer and n is any nonnegative integer, let rs(8n + s) be the number of solutions of the equation
$$x_1^2 + x_2^2 + ... + x_s^2 = 8n + s$$
in integers x1, x2,…, xs, and let rs*(8n + s) be the number of solutions of the same equation in odd integers. Alternatively, rs*(8n + s) is the number of ways of expressing n as a sum of triangular numbers, i.e., the number of solutions of the equation
$$\frac{{{y_1}({y_1} - 1)}}{2} + \frac{{{y_2}({y_2} - 1)}}{2} + ... + \frac{{{y_s}({y_s} - 1)}}{2} = n$$
in integers yl, y2,…,ys. It is known that for 1 ≤ s ≤ 7 there exists a positive constant cs such that
$${r_s}(8n + s) = {c_s}{r_s}^*(8n + s)$$
for all nonnegative integers n. In this paper we prove that if s > 7, then no constant csexists such that (*) holds, even for all sufficiently large n. The proof uses the theory of modular forms of weight s/2 and appropriate multiplier system on the group Γ0(64) and the so-called principle of the preservation of modularity under congruence restrictions.

Keywords

Sums of squares modular forms preservation of modularity under congruence restrictions 

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Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Paul T. Bateman
    • 1
  • Boris A. Datskovsky
    • 2
  • Marvin I. Knopp
    • 2
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA
  2. 2.Department of MathematicsTemple UniversityPhiladelphiaUSA

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