With a graduate student, I'm going through the paper (Proc. London Math. Soc. (3) 47 (1983), no. 2, 193–224.) 


Here's the background and notation. 


We have a quadratic character $\chi$ modulo $q$, with Siegel zero $\beta\_0$. $\eta=((1-\beta\_0)\log q)^{-1}$, so $3\le \eta\ll q$ is known. Let $L=\log q$. We take
$$
q^{250}\le x\le q^{500}.
$$


Lemma 3 (stated on p. 198) says
$$
\sum\_{p<x,\chi(p)=1}\frac{\log(p)}{p}\ll L\log(\eta)^{-1/2}
$$
Heath-Brown says "This is not necessarily the best bound of its type, but suffices for our purposes." I think the proof he gives is flawed. I know results like this are found elsewhere. My question is


**Can this proof be fixed? If not, can you provide a reference for a more robust proof?**


(Side note: In a recent blog post on this same paper, Tao proves his Lemma 5 which is similar, summing instead $1/p$, up to a $o(1)$ error. "For more precise estimates on the $o(1)$ error, see the paper of Heath-Brown (particularly Lemma 3).")


The proof starts in Section 4 on p. 206. The overall structure is to write 
$$
\frac{L^\prime}{L}(s,\chi)-\frac{L^\prime}{L}(s^\prime,\chi)
$$
as a both sum over zeros and as a sum over primes. Here $s=1+L^{-1}$ and $s^\prime=1+aL^{-1}$, where $a$ is to be chosen later. The zeros side comes down to estimating
$$
aL^{-1}\sum\_{\rho\ne\beta\_0}|\rho-1|^{-2}.
$$
The zeros $\rho$ at height $\ge 1$ give no trouble. Heath-Brown quotes Prachar to estimate the number of zeros in the disk $|s-1|\le r$ as
$$
\ll 1+r\log q
$$ 
for $r\le 2$. To count zeros below height $1$ it would make sense to take $r=\sqrt{2}$ here.


With $r\_0$ the closest non-Siegel zero to $1$, he quotes the Deuring-Heilbronn phenomenon to say $r\_0\gg L^{-1}\log\eta$.


One would expect then the bound to involve the $aL^{-1}$ term, the number of zeros at height below $1$, and the worst case for the term being summed, namely $r\_0^{-2}\ll L^2\log(\eta)^{-2}$, i.e. a bound of
$$
aL(1+\sqrt{2}L)\log(\eta)^{-2}\ll aL^2\log(\eta)^{-2}.
$$
Heath-Brown uses instead the count of zeros inside the circle of radius $r\_0$ (which makes no sense, by definition of $r\_0$) and gets a better estimate $\ll aL/\log\eta$. The correct (I think) estimate does not suffice for the error bound the lemma claims.



Another question on Heath-Brown's "Prime twins and Siegel zeros"
"Those oft are stratagems which errors seem, 
Nor is it Homer nods, but we that Dream."


Heath-Brown's proof is fine. It needs just one more line of explanation.  

Note that 
$$
\sum\_{\substack{\rho \neq \beta \\ |\gamma| \le 1}} \frac{1}{|\rho -1|^2} \ll 
\int\_{r\_0}^{2} \# \{ \rho \neq \beta: |\rho -1| \le x \} \frac{dx}{x^3}, 
$$  

and by the quoted line from Prachar this is 
$$ 
\ll \int\_{r\_0}^2 (1+x \log q) \frac{dx}{x^3} \ll r\_0^{-2} + (\log q) r\_0^{-1}, 
$$ 
which is what Heath-Brown writes. 


