Question regarding proof of Mertens’ estimates in Montgomery-Vaughan’s “Multiplicative number theory”

I have been trying to read Theorem 2.7 of Montgomery-Vaughan’s “Multiplicative number theory” volume 1, and there is an issue I have run into: in the proof of subpart (e), when they write
$$delta int_1^infty frac{, dx}{x^{1+delta}} sum_{n leq x} frac{Lambda(n)}{n log n} = sum_{n geq 1} frac{Lambda(n)}{n^{1+delta} log n} = log zeta(1+delta),$$
they refer to the Corollaries 1.3, 1.11 and 1.13. From what I can see, they want to apply Corollary 1.3 to the Dirichlet Series corresponding to the sequence $(a_n)_{n geq 1}$ given by $a_n:= Lambda(n) / (n log n)$. This Dirichlet series is
$$sum_{n geq 1} frac{Lambda(n)}{n^{1+s}log n} = log zeta(s+1) hspace{5mm} text{ for } Re(s)>0$$
and is clearly seen to have abscissa of absolute convergence (hence also abscissa of convergence) at most $0$ (and I think equality holds, which is basically the root of my problem). However in Corollary 1.3 (in the part they seem to be using in order to get the aforementioned identity) applies only when the abscissa of convergence is strictly negative. Even using Corollary 1.13 (which is the result stating the meromorphic continuation of the Riemann Zeta to the half plane $sigma>0$ with simple pole of residue $1$ at $s=1$) doesn’t seem to be useful – as $log zeta(s+1)$ then has a singularity at $s=0$, so it seems to give a result of the same strength as the one given by our absolute convergence argument above. I would really like to know what I am (probably unnecessarily) tearing my hair out over. Thanks.