Representations of Integers as Sums of Squares

Representations of integers as sums of an even number of squares

“…This conjecture was also proved independently in a different way, using the theory of modular forms, by Zagier [14]. In this connection see also the work of Ono [10] where the arguments used are slightly different.…”

“…are that for r s (n) with s = 4j 2 and resp., s = 4j 2 + 4j he needs only j such functions, and that in his corresponding Eisenstein series expansions of θ to these powers, the maximum weight of any Eisenstein series that appears is 4j − 2 and 4j , respectively. The sums of squares and triangular numbers formulas, as well as the corresponding powers of theta series expansions in [4], [10], [12], [14] also all have this same maximum weight of their Eisenstein series. Even though the maximum weight of the Eisenstein series in our expansions is half the power of the theta functions involved, our results seem to be rather optimal from another standpoint.…”

On special values of certain Dirichlet L-functions

“…The study of r k (n) has a long history. When k is even several formulas are available for r k (n) (for details, see [6][7][8][9]). It is worthwhile to note that a well known theorem of Gauss relates r 3 (n) to class numbers of imaginary quadratic fields.…”

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

“…for every positive integer s. Using Zagier's work [17] on the Kac-Wakimoto conjecture, K. Ono [11] obtained formulas (simpler than Milne's) for r 4s 2 (n) and r 4s 2 +4s (n), which are sums of products of divisor functions. For odd values of k a general formula is not known, though formulas for certain values of k are known.…”

“…For odd values of k a general formula is not known, though formulas for certain values of k are known. For example, the formulas for r k (n) or r k (n 2 ) are known for k = 1, 3,5,7,9,11,13. See [2,4,5,7,8,[14][15][16] for details.…”