Exponential decay in the square of a logarithm

Statement

For fixed $c>0$ , $A,B\ge0$ , and every $N\ge0$ ,

q^{-A}(1+|\log q|)^B e^{-c(\log(1/q))^2}=O(q^N) \qquad(q\downarrow0).

Thus decay exponential in the square of the logarithm is stronger than any fixed power of $q$ , even after polynomial and logarithmic losses.

Proof and derivative use

Set $s=\log(1/q)$ . Dividing the left side by $q^N$ gives $(1+s)^B\exp(-cs^2+(A+N)s)$ , which tends to zero since the negative quadratic term dominates. If each prescribed derivative of a family obeys a bound of this form, with its own fixed $A,B,C$ , every derivative has infinite-order decay. The scalar value estimate alone makes no claim about derivatives of another family sharing that value bound.