Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 1 / c / Solution

For $t>0$, the product from part (b) has $q=e^{-2\pi t}\in (0,1)$, soThe weight-twelve transformation law givesTogether with exponential decay as $t\to \infty$, this implies rapid decay at both endpoints for the Mellin transformThe integral therefore converges for every real $s$, and its integrand is strictly positive.

In the half-plane where the Dirichlet series may be integrated term by term,Analytic continuation preserves this identity. For real $s>0$, both $(2\pi {)}^s$ and the Gamma function $\Gamma (s)$ are positive, soThis is the positivity statement in the Mellin transform of the modular discriminant, and in particular $L(\Delta ,s)\ne 0$.