I'm stuck on a line in the proof of Bombieri implies Linnik, where
Bombieri: For primitive $\chi$ mod $q$ with $q \leq T$ we define $$N(\alpha, T; \chi)=\#\{\rho=\beta+i\gamma \;:\; \Lambda(\rho,\chi)=0, \beta>\alpha, |\gamma|<T\}.$$
There exists $c>0$ such that, for all $T \geq 2$ and all $1/2 \leq \alpha <1$, $$\sum_{q \leq T}\sum_{\chi \text{ mod }q}^*N(\alpha,T;\chi)=O(T^{c(1-\alpha)})$$
where the asterisk on the sum over characters indicates that only primitive characters are to be inlcuded.
If there exists a real primitive character $\chi_1$ mod $q_1$ with $q_1<T$, such that $\Lambda(s,\chi_1)$ has zero $\beta_1>1-\frac{\delta}{\log T}$, then we have the improved estimate
$$\sum_{q \leq T}\sum_{\chi \text{ mod }q}^*N'(\alpha,T;\chi)=O(T^{c(1-\alpha)}(1-\beta_1)\log T)$$
where the ' on the $N(\alpha,T;\chi)$ indicates that the single exceptional zero is to be excluded.
Linnik: If $a$ and $q$ are coprime then there exists a prime $p \equiv a$ mod $q$ with $p<O(q^{O(1)})$.
In the exceptional zero case, assuming Bombieri, I've got to the point where
$$\sum_{p \leq x, p \equiv a \text{ mod }q }\log p \geq \frac{1}{2 \phi(q)}\left(x-\frac{x^{\beta_1}}{\beta_1}\right)+O\left(\frac{x(\log x)^2}{T}\right)$$
for $T=q^A$ with $A>3$ and $x=T^B$ with $B>12$ a large constant. I also know that $$\beta_1\leq 1-\frac{c}{\sqrt{T}(\log T)^2}.$$
I think what I want to show is that $x^{\beta_1}\leq x^{1-\epsilon}$ for some $\epsilon>0$, because since $(O(x(\log x)^2/T)=O(x^{1-\epsilon})$ for small enough $\epsilon$, Linnik's theorem will follow. But just throwing in the bound for $\beta_1$ doesn't seem to give me this.
We may write $$x - \frac{x^{\beta_1}}{\beta_1} = x\left(1 - \frac{x^{\beta_1 - 1}}{\beta_1}\right).$$
Consider two cases. In the first case, $\beta_1 < 1 - \frac{c}{\log x}$ for a fixed constant $c$. In this case the answer is $\gg x$, which is more than sufficient. Otherwise, note that for $t \in [1 - \frac{c}{\log x}, 1]$ we have $$\frac{d}{dt} \frac{x^{t-1}}{t} = \frac{x^{t-1}\log x}{t} - \frac{x^{t-1}}{t^2} \gg \log x.$$ Therefore, by the mean value theorem, $$ x\left(1 - \frac{x^{\beta_1 - 1}}{\beta_1}\right) \gg x(1-\beta_1) \log x.$$ Again, this bound suffices, since $q \leq T^{\frac{1}{3}}.$