Preliminary Facts Sample Clauses

Preliminary Facts. Consider the generating function f associated to Qd(n), f (τ ) := Y (1 − qn)−1 = 1 + Σ Qd(n)qn, n≡±1(d+3) n≥0 where q = e−τ and R(τ ) > 0. Let τ = y + 2πix. We can then obtain a formula for Qd(n) by integrating f (τ ) against enτ . Consequently, we require an approximation of f (τ ) so that we may make use of this integral formula. To do this, we need an xxxxxxxxxx xxxxxxxx, X x(x ) := n≡±1(d+3) n≥0

Preliminary Facts. For fixed d ≥ 4, we have the generating function Σ f (z) := qd(n)e−nz (4.11) with z = x + iy. Hence, we obviously have that qd(n) = f (z)xxx xx. (4.12) 1 ∫ π 2π Therefore to estimate qd(n) we require strong approximations for f (z).

Preliminary Facts. Σ We begin by recalling some basic facts about modular forms. A holomorphic function f : H → C is a weakly holomorphic modular form of weight k ∈ 1 Z for the subgroup Γ ⊆ SL2(Z) if f (γz) = sγ(cz + d)kf (z) for all γ =