Representations of integers as sums of an even number of squares

be the standard theta function. Then θ which is the generating function for rs(n) (n ∈ N), is a modular form of weight s 2 and level 4 and hence for s ≥ 4 can be expressed as the sum of a modular form in the space generated by Eisenstein series of weight s 2 and level 4 and a corresponding cusp form. Recall that the Fourier coefficients of Eisenstein series of integral weight ≥ 2 on congruence subgroups are given explicitly in terms of elementary divisor functions. Thus if the space of cusp forms happens to be zero, for even s ≥ 4 this leads to exact formulas for rs(n) in terms of those functions. In general, however, one gets only asymptotic results for rs(n) when n is large, since the coefficients of cusp forms are mysterious and in general no simple arithmetic expressions are known for them.