My question is just a ``I don't understand what goes on in X of paper Y" and is maybe a bit long. I don't like to post it on overflow cause it seems to be too much of an "can anyone explain this in this paper", but it is still a research question so if I can't find an answer here I'll post it there.
So my question is on Christopher Hooley's paper on the third moment for primes in arithmetic progressions (``On the Barban-Davenport-Halberstam Theorem VIII"):
For variables $z,z_1,\Delta $ he defines two quantities, $\mathcal J_1(z,\Delta )$ and $\mathcal J_2(z_1,\Delta )$, in terms of quantities $\Gamma _\Delta (\cdot )$. As far as my question is concerned, I don't think we don't need to know anything about these $\Gamma _\Delta (\cdot )$. The quantities $\mathcal J_1(z,\Delta )$ and $\mathcal J_2(z_1,\Delta )$ are as follows. From (64) we have \[ \mathcal J_1(z,\Delta )=\sum _{l<z}\frac {(z-l)^2}{l}\Gamma _\Delta (l) \sum _{l_1+l_2=l}\Gamma _\Delta (l_1)\Gamma _\Delta (l_2)\] and from (68) we have \[ \mathcal J_2(z_1,\Delta )=\sum _{l<z_1}(z_1-l)^2l\Gamma _\Delta (l) \sum _{l_1+l_2=l}\Gamma _\Delta (l_1)\Gamma _\Delta (l_2).\] From (112) we have \[ (1)\hspace {10mm}\mathcal J_2(z_1,\Delta )=\text { main term } +\mathcal O\left (\frac {\Delta ^\epsilon z_1^{7/2}}{z^{1/4}}\right ) +\mathcal O^z\left (\Delta ^\epsilon z_1^{7/2}e^{-\sqrt {\log (z_1+2)}}\right )\] for any $z$ with $z\geq z_1$, according to the top of page 38. Here the symbol $\mathcal O^z(f)$ denotes a quantity that is $\mathcal O(f)$ and independent of $z$, also said at the top of page 38.
(This introduction of a new variable is described through the expression \[ \mathcal J_2(z_1,\Delta )=\sum _{d<z\atop {(d,2\Delta )=1}}\frac {\mu ^2(d)}{\theta _2(d)} \sum _{l<z_1\atop {l\equiv 0\text { mod }d}}(z_1-l)^2l\sum _{l_1+l_2=l}\Gamma _\Delta (l_1)\Gamma _\Delta (l_2)\] which is the equality immediately after (68), and through the bottom of page 37, which says we can change the $d$ summation range from $d<z_1$ to $d<z$ for any $z\geq z_1$, since the condition $d<z_1$ is automatic from the $l\equiv 0\text { mod }d$ condition in the $l$ sum.)
From Lemma 3 (page 25) $\mathcal J_1(z,\Delta )$ and $\mathcal J_2(z_1,\Delta )$ are related through \[ (2)\hspace {10mm}\mathcal J_1(z,\Delta )=\frac {\mathcal J_2(z,\Delta )}{z^2}-6z\int _{0}^z\frac {\mathcal J_2(z_1,\Delta )dz_1}{z_1^4} +12z^2\int _{0}^z\frac {\mathcal J_2(z_1,\Delta )dz_1}{z_1^5}.\] At the bottom of page 38 and the first half of page 39 he says he inserts $(1)$ into $(2)$ to calculate $\mathcal J_1(z,\Delta )$ up to an error \[ \mathcal O\left (\Delta ^\epsilon z^{3/2}e^{-A'\sqrt {\log (z+2)}}\right ) +\mathcal O\left (\Delta ^\epsilon z\int _1^z\frac {e^{-A'\sqrt {\log (u+2)}}}{u^{1/2}}du\right )\] \[ 12z^2\int _1^\infty \mathcal O^z\left (\frac {\Delta ^\epsilon e^{-A'\sqrt {\log (u+2)}}}{u^{3/2}}\right )du +\mathcal O\left (z^2\Delta ^\epsilon \int _z^\infty \frac {e^{-A'\sqrt {\log (u+2)}}}{u^{3/2}}du\right )+O\left (z^{7/4}\Delta ^\epsilon \int _1^\infty \frac {du}{u^{3/2}}\right ).\] He then says (in the same equation passage) the total error above is \[ z^{3/2}\Delta ^\epsilon e^{-A'\sqrt {\log (z+2)}}\] which to me doesn't seem to be right in view of the $12z^2$ term.
My first question is simple - is this a mistake/slip?
If not, how does it follow? If it is a mistake, what could he have meant? I've tried to work out what he could have meant, but as far as I can see, even if $(1)$ read more strongly \[ \mathcal J_2(z_1,\Delta )=\text { main term }+\mathcal O\left (1\right )\] and we inserted this into $(2)$ we couldn't avoid an error term of the form \[ 12z^2\int _{0}^z\frac {dz_1}{z_1^5}\] so that we can't calculate $\mathcal J_1(z,\Delta )$ up to an error better than $z^2$.
But this would mean the whole result is wrong, since the main term is of this size, so my interpretation/understanding is clearly nonsense. So my second question is: can anyone clear this up from me?