10 1. Introduction The formulas for Sk(n) given above are derived using the theory of modular forms, as follows. We define a formal power series Θ(q) by Θ(q) = j=−∞ qj2 and, for k 2, consider the power series expansion of the kth power of Θ: (Θ(q))k = j=−∞ qj2⎠ ⎞k = a1=−∞ qa1 2 · · · ak=−∞ qak 2 = n≥0 cnqn. What is the nth coefficient, cn, of Θk? If the readers stare at the previous equation for a while, they will find that cn is given by cn = #{(a1,...,ak) Zk : a2 1 + · · · + a2 k = n}. Therefore, cn = Sk(n) and (Θ(q))k = n≥0 Sk(n)qn. In other words, Θk is a generating function for Sk(n). But, how do we find closed formulas for Sk(n)? This is where the theory of modular forms be- comes particularly useful, for it provides an alternative description of the coefficients of Θk. It turns out that, for even k 2, the function Θk is a modular form of weight k 2 (more precisely, it is a modular form for the group Γ1(4)), and the space of all modular forms of weight k 2 , denoted by M k 2 (Γ1(4)), is finite dimensional (we will carefully define all these terms later). For instance, let k = 4. Then M2(Γ1(4)), the space of modular forms of weight 4 2 = 2 for Γ1(4), is a 2-dimensional C-vector space and a basis is given by modular forms with q-expansions: f(q) = 1 + 24q2 + 24q4 + 96q6 + 24q8 + 144q10 + 96q12 + · · · g(q) = q + 4q3 + 6q5 + 8q7 + 13q9 + 12q11 + 14q13 + · · · .
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