On the representation of integers as sums of an odd number of squares

Abstract

In a recent work, S. Cooper (J. Number Theory 103:135-162, 1988) conjectured a formula for r 2k+1 (p 2 ), the number of ways p 2 can be expressed as a sum of 2k + 1 squares. Inspired by this conjecture, we obtain an explicit formula for r 2k+1 (n 2 ), n ≥ 1.A classical problem in number theory is to give an explicit formula for the number of ways one can represent a non-negative integer n as a sum of k squares, where k is a positive integer. The study of r k (n) has a long history. For k = 2, 4, 6, 8, elegant formulae for r k (n) were found by Jacobi. A general formula for r k (n), when k is even was stated by Ramanujan [12]. It was proved by Mordell [10] and S. Cooper [3] gave an elementary proof using Ramanujan's 1 ψ 1 summation formula. It is also known that r k (n) can be expressed in terms of coefficients of Eisenstein series and cusp forms and Rankin [13] showed that the cusp form part is non-trivial for k > 8. Recently, combining a variety of methods, S. Milne [9] obtained formulas for r 4s 2 (n) and r 4s 2 +4s (n) Dedicated to Srinivasa Ramanujan.S. Gun · B. Ramakrishnan ( )

A canonical subspace of modular forms of half-integral weight

“…3 We now make some remarks about the question of Zagier. In [5], the first and last authors constructed a basis of M 2k ( 0 (2)) using the eta-quotients as follows [5, Lemma 3.1]: Let k be an even integer. For j = 1, 2, .…”

“…In [5], the first and the last authors constructed a basis for M 2k ( 0 (2)) and obtained a formula for r 2k+1 (n 2 ), the number of representations of n 2 as a sum of 2k + 1 squares. They also made some observations through examples and in this connection, in a private discussion with the first author, Zagier predicted the existence of a canonical subspace of M k+1/2 ( 0 (4)), different from the Kohnen + space, which is mapped to M 2k (which is the vector space of modular forms of weight 2k for the full modular group SL 2 (Z)) under the Shimura map S 1,2 (see Remark 3.3 for details).…”

On special values of certain Dirichlet L-functions

“…In our earlier paper [3] we have used this method to get a general formula for r 2k+1 (n 2 ). For an even integer k ≥ 4, let…”

“…It is worthwhile to note that a well known theorem of Gauss relates r 3 (n) to class numbers of imaginary quadratic fields. However, there is no explicit formula known for a general odd integer k. In our earlier work [3], we gave a formula for r 2k+1 (n 2 ) using the Shimura correspondence. In [1], Shaun Cooper announced two conjectures regarding the values of r 9 (t) and r 11 (t) for certain class of square-free positive integers.…”

A NOTE ON SPECIAL VALUES OF CERTAIN DIRICHLET L-FUNCTIONS

“…In our earlier work with Gun [6], we proved the conjectures of Cooper [3] on certain formulas for r 9 (m 1 ) and r 11 (m 2 ), m 1 , m 2 square-free integers with m 1 ≡ 5 mod 8, m 2 ≡ 7 mod 8, expressing them as finite sums involving Jacobi symbols. Then, using the formula for r k (n) obtained in [7], we derived the expressions for the special values of the L-functions, namely, L(−3, χ m 1 ) and L(−4, χ −m 2 ) in terms of finite sums involving Jacobi symbols.…”