1.2. Modular forms 9 and S2(9) = 4. Indeed, the number nine has 4 different representa- tions: 9 = (±3)2 + 02 = 02 + (±3)2. Let us explore other similar questions. Let n 0 and k ≥ 2. Is the number n 0 a sum of k (integer) squares? In other words, are there a1,...,ak ∈ Z such that n = a1+···+ak? 2 2 And if so, in how many different ways can you represent n as a sum of k squares? Lagrange showed that every natural number can be represented as a sum of k ≥ 4 squares, but how many different representations are there? Let Sk(n) be the number of representations of n as a sum of k squares. Determining exact formulas for Sk(n) is a classical problem in number theory. There are exact formulas known in a number of cases (e.g. Eq. 1.2). The formulas for k = 4,6 and 8 are due to Jacobi and Siegel. We write n = 2νg, with ν ≥ 0 and odd g 0: S4(n) = 8 d|n, 4 d d, S6(n) = −1 g 22ν+4 − 4 d|g −1 d d2, S8(n) = 16 · ∑ d3 if n is odd, ∑d|n d|n d3 − 2 ∑ d|g d3 if n is even. For example, S4(4) = 8(1 + 2) = 24 and, indeed 4 = (±1)2 + (±1)2 + (±1)2 + (±1)2 = (±2)2 + 0 + 0 + 0 = 0 + (±2)2 + 0 + 0 = 0 + 0 + (±2)2 + 0 = 0 + 0 + 0 + (±2)2. So there are 16 + 2 + 2 + 2 + 2 = 24 possible representations of the number 4 as a sum of 4 squares. Notice that S4(2) = S4(4). In how many ways can 4 be represented as a sum of 6 squares? We write 4 = 22 · 1, so ν = 2 and g = 1, and thus, S6(4) = −1 1 22·2+4 − 4 −1 1 · 12 = (28 − 4) · 1 = 252.

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