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)?



A simple question about the classical divisor problems
It is known (see Titchmarsh Chapter 12) that if you define $\gamma\_k$ the lower bound of $\sigma > 0, \int\_{-\infty}^{\infty}\frac{|\zeta(\sigma+it)|^{2k}}{|\sigma+it|^2}dt < \infty$, then $\frac{k-1}{2k}\le \gamma\_k=\beta\_k \le \alpha\_k$, where $\beta\_k$ is the usual mean bound of $\Delta\_k$ ( lowest bound of orders for which $\frac{1}{x}\int\_{0}^{x}{\Delta\_k(y)^2}dy=O\_{\epsilon}(x^{2\beta\_k+\epsilon})$) and the integral above in the question converges to $\Delta\_k(x)$ for $\gamma\_k< \gamma <1$. 


Since it is known that for example $\gamma\_2=\frac{1}{4}, \gamma\_3=\frac{1}{3}, \gamma\_4=\frac{3}{8}$ (best possible, first two results appearing in Titchmarsh, last in Ivic, chaper 13), while the known respective values for $\alpha$ are still far away from those, even in the simplest cases $k=2,3,4$ the equivalence required is far from being proven and it is equivalent to fully solving the Dirichlet Divisor problem for $k=2,3,4$. 


In general, obviously, even less is known about $\gamma\_k, \alpha\_k$ so the question is on par with Lindelof in many ways (as Lindelof is equivalent to any and all of $\alpha\_k \le \frac{1}{2}, \gamma\_k \le \frac{1}{2}, \gamma\_k = \frac{k-1}{2k}$)


