In the classical divisor problems, for $k\geq 2$, $\alpha_k$ usually denotes the infimum of real numbers $\sigma<1$ such that $$\Delta_k(x)=\sum_{n\leq x}d_k(n)-\textrm{Res}\left(\frac{\zeta^k(z)x^z}{z},z=1\right)=O(x^{\sigma})$$ as $x\rightarrow\infty$.

As often happens in analytic number theory, one might expect that $\alpha_k$ is also the infimum of $\sigma<1$ such that the integral $$\lim_{T\rightarrow\infty}\int_{\sigma-iT}^{\sigma+iT}\frac{\zeta^k(s)x^sds}{s}$$ exists for all $x>0$.

However, it appears to me that this equivalence is not so easily proved, nor can I find a proof in the literature. Therefore, I would like to ask if this is actually a known theorem or, if not, whether the difference has been been investigated and in which paper(s)?