Celebratio Mathematica — Bateman

Sums of squares and modular number theory

Works connected to Francis Gerard Garvan

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P. T. Bateman, B. A. Datskovsky, and M. I. Knopp: “Sums of squares and the preservation of modularity under congruence restrictions,” pp. 59–71 in Symbolic computation, number theory, special functions, physics and combinatorics (Gainesville, FL, 11–13 November 1999). Edited by F. G. Garvan and M. E. H. Ismail. Developments in Mathematics 4. Kluwer Academic (Dordrecht), 2001. MR 1880079 Zbl 1040.11018 incollection

If \( s \) is a fixed positive integer and \( n \) is any nonnegative integer, let \( r_s(8n + s) \) be the number of solutions of the equation \[ x_1^2 + x_2^2 + \cdots + x_2^s = 8n + s \] in integers \( x_1, x_2, \dots, x_s \), and let \( r_s^*(8n + s) \) be the number of solutions of the same equation in odd integers. Alternatively, \( r_s^*(8n + s) \) is the number of ways of expressing \( n \) as a sum of triangular numbers, i.e., the number of solutions of the equation \[ \frac{y_1(y_1 - 1)}{2} + \frac{y_2(y_2 - 1)}{2} + \cdots +\frac{y_s(y_s - 1)}{2} = n \] in integers \( y_1, y_2,\dots, y_s \). It is known that for \( 1 \leq s \leq 7 \) there exists a positive constant \( c_s \) such that \begin{equation*}\tag{*} r_s(8n + s) = c_sr_s^*(8n + s) \end{equation*} for all nonnegative integers \( n \). In this paper we prove that if \( s > 7 \), then no constant \( c_s \) exists 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 \( \Gamma_0(64) \) and the so-called principle of the preservation of modularity under congruence restrictions.

Marvin Isadore Knopp	Related
Mourad El Houssieny Ismail	Related
Francis Gerard Garvan	Related
Boris Abramovich Datskovsky	Related