For sure, we can write \begin{align} \zeta(s)&=\int_0^\infty \sum_{n\geq 1}\delta\big(\log \frac{x}{n}\big)x^{-s-1}dx\\ &=\int_0^\infty \sum_{n\geq 1}\delta(x-n)x^{-s}dx \nonumber \\ &=\sum_{n\geq 1}n^{-s}\quad \forall s | \Re\{s\}>1 \nonumber \\ \end{align} where $\delta$ is Dirac's delta.
In an answer by reuns here, I find that we add a regularization term for $0<\Re\{s\}<1$ : $$\zeta(s)=\int_0^\infty \Big(-x+\sum_{n\geq 1}\delta\big(\log \frac{x}{n}\big)\Big)x^{-s-1}dx\quad $$
My question is: Is the first expression sufficient in the case of a variable $s$ with a non-vanishing imaginary part $\Im\{s\}\neq 0$ ?
It seems to me that the oscillatory behaviour introduced by the imaginary part of $s$ may allow, in the first expression, the exchange of the sum and the integral, and the convergence of the series, although it is not an absolute convergence for $\Re<1$. Is it correct?