The cutoff $\eta_0$ is nonnegative

Tao's logarithmic cutoff $\eta_0(t) = 4(\log 2 - |\log 2t|)_+$, extended by zero to $t \leq 0$, is nonnegative everywhere. This is immediate from its definition as four times a maximum with zero, but it is worth having available: $\eta_0$ appears as a weight in the third prime sum of equation (8.10) and in the exponential sum $S_{\eta_0,q_0}$ of Theorem 1.3, and nonnegativity of the weights is what makes the representation count nonnegative and hence makes its positivity equivalent to the existence of a representation.