Celebratio Mathematica

P. T. Bateman: “Integers expressible in a given number of ways as a sum of two squares,” pp. 37–45 in A tribute to Emil Grosswald: Number theory and related analysis. Edited by M. I. Knopp and M. Sheingorn. Contemporary Mathematics 143. American Mathematical Society (Providence, RI), 1993. Dedicated to the memory of Elizabeth and Emil Grosswald. MR 1210510 Zbl 0790.11032 incollection

Marvin Isadore Knopp	Related
Mark Sheingorn	Related
Emil Grosswald	Related
Elizabeth Grosswald	Related

P. T. Bateman and M. I. Knopp: “Some new old-fashioned modular identities,” Ramanujan J. 2 : 1–2 (1998), pp. 247–269. Dedicated to the memory of Paul Erdős. MR 1642881 Zbl 0909.11018 article

This paper uses modular functions on the theta group to derive an exact formula for the sum \[ \sum_{|j|\leq n^{1/2}}\sigma(n - j^2) \] in terms of the singular series for the number of representations of an integer as a sum of five squares. (Here \( \sigma(k) \) denotes the sum of the divisors of \( k \) if \( k \) is a positive integer and \( \sigma(0) = -1/24 \).)

Several related identities are derived and discussed.

Two devices are used in the proofs. The first device establishes the equality of two expressions, neither of which is a modular form, by showing that the square of their difference is a modular form. The second device shows that a certain modular function is identically zero by noting that it has more zeros than poles in a fundamental region.

Marvin Isadore Knopp	Related
Paul Erdős	Related

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

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Paul T. Bateman

Sums of squares and modular number theory

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