The book of Titchmarsh and Heath-Brown on the Riemann zeta function talks about the integral ($c > 0$) $$ \frac{1}{2\pi i} \left( \int_{-\infty - iT}^{c - iT} + \int_{c - iT}^{c+iT} + \int_{c + iT}^{-\infty + iT} \right) \left( \frac{x}{n} \right)^w \frac{dw}{w}, $$ and I swear to god that they consider the case $n < x$, stating that a "calculus of residues" would imply that the integral equals $1$. (It does equal one on a finite domain containing 0 by the well-known residue theorem.) This is p. 60, lemma 3.12.
But if we rewrite the integrand as $$ \frac{1}{w} \exp\left( \ln\left( \frac{x}{n} \right) w \right), $$ it becomes clear that the integrand (and hence the integral) diverge in the worst possible fashion as $\Re w \to \infty$. What is going on here? How to even interpret the integral?
As noted in the comments, the real part goes to $-\infty$, so that the integral does converge. I misread the domain of integration.