The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c} a\text{ mod q}\\ (a,q)=1 \end{array}}\left|\psi(y;q,a)-\frac{y}{\phi(q)}\right|\ll \frac{x}{(\log x)^A}.$$

I was wondering what happens if a restriction is put on the $q$ so that they are all divisible by some smaller integer $k$. Are there any non-trivial bounds on the average over $q$ divisible by $k$? Specifically, suppose that $k\leq Q^{1-\epsilon},$ and that there is no Siegel zero${}^{++}$ for any $\chi$ modulo $k$. Is it true that $$\sum_{\begin{array}{c} q\leq Q\\ k|q \end{array}}\max_{y\leq x}\max_{\begin{array}{c} a\text{ mod q}\\ (a,q)=1 \end{array}}\left|\psi(y;q,a)-\frac{y}{\phi(q)}\right|\ll_{\epsilon}\frac{1}{k} \frac{x}{(\log x)^A}.$$

Any references would be greatly appreciated.

Thanks for your help,

${}^{++}$ Edit: As mentioned by Terence Tao in the comments, there is an issue regarding Siegel zeros modulo $k$. The original result I was asking about would give stronger bounds on the location of a possible Siegel zero modulo $k$, and for that reason it is out of reach.

I worked out a sketch/heuristic, as suggested by Terence Tao's comment, for why we have to assume that there are no Siegel zeros modulo $k$. I wrote this mainly for my own understanding, and I have added it here for anyone who is interested. I would still like to know if the result holds assuming there are no non-trivial real zeros for $\chi$ mod $k$.

Heuristic: Suppose that $k$ is a small power of $x$. If there is an exceptional zero $\beta$ for a quadratic character $\chi$ modulo $k$ , then for every $q$ such that $k|q,$ the induced character modulo $q,$ $\chi^{\star},$ will have the same exceptional zero, and so we expect that $$\psi(x;q,a)\approx\frac{x}{\phi(q)}-\frac{\chi^{\star}(a)}{\phi(q)}\frac{x^{\beta}}{\beta}+small$$ for each $q.$ This leads us to expect that $$\sum_{\begin{array}{c} q\leq Q\\ k|q \end{array}}\max_{y\leq x}\max_{a\text{ mod }q}\left|\psi\left(y;q,a\right)-\frac{y}{\phi(q)}\right|\approx x^{\beta}\sum_{\begin{array}{c} q\leq Q\\ k|q \end{array}}\frac{1}{\phi(q)}\approx\frac{x^{\beta}\log x}{k},$$ however, the desired upper bound is too strong, as $$x^{\beta}\log x\ll \frac{x}{\left(\log x\right)^{B}},$$ implies that for some $C>0$, $$\beta\leq1-C\frac{\log\log x}{\log x},$$ which is stronger than the long standing bound by Siegel $$\beta\leq1-C(\epsilon)x^{-\epsilon}.$$ In the above, we have used $x$ interchangeably with $k$ for the bounds since $k$ is a small power of $x$.

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    $\begingroup$ I think you want $Q \leq\sqrt{x}/\log^B x$ rather than $\sqrt{x}/\log^B x \leq Q \leq \sqrt{x}$. The bound you want is true on GRH, but I'd be surprised if one can get something this strong unconditionally. In particular, a Siegel zero at the quadratic character with conductor k is likely to be inconsistent with this claim (though I haven't checked this carefully). $\endgroup$
    – Terry Tao
    Jul 16 '13 at 17:39
  • $\begingroup$ @TerryTao: Thanks, I have now corrected the statement of the Theorem. I'll try and work out the contributions from the Siegel zero. $\endgroup$ Jul 16 '13 at 17:52
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    $\begingroup$ What you need follows by zero density estimates, assuming that there are no Siegel zeroes for primitive characters mod $km$ with $m\le (\log x)^B$. Look up the Grand Density Theorem (thm 10.4 in Iwaniec-Kowalski). This should give you the result when $k\le x^{\epsilon/\log\log x}$ for an appropriate $\epsilon$. It might be possible to improve this range using the log-free zero density estimate (eq. (18.9) in Iwaniec-Kowalski). $\endgroup$ Jul 19 '13 at 19:51
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    $\begingroup$ I should also add that in order to use the Grand Density Theorem, you need to remove the "bad" moduli. If $k$ is prime, then "bad" means that either $d>(\log x)^B$ or $d=km$ with $m>(\log x)^B$, and $N(1-M\log\log x/\log x,\sigma,\chi,x^3)>0$ for an appropriate constant $M$. $\endgroup$ Jul 19 '13 at 19:55

Elliott has a result in this direction. Let $A>0$ and $a \geq 2$ a fixed integer. Furthermore, let $q \leq x^{1/3-\epsilon} $ be a large power of $a$. One then has that $$ \sum_{\substack{d \leq q^{-1}x^{1/2} \log^{-A-6}(x) \\ (d,q)=1 }} \max_{(r,qd)=1} \max_{y\leq x} \left|\pi(y,qd,r) - \frac{\text{Li}(y)}{\phi(qd)} \right| \ll_{A} \frac{x}{\phi(q) \log^{A}(x) } .$$ Taking $d=1$ recovers the Bombieri-Vinogradov theorem. Note that while Bombieri-Vinogradov doesn't contain any information about specific arithmetic progressions with modulus larger than $\log^{A}(x)$, Elliott's inequality implies that $$\pi(x,q,r) \sim \frac{Li(x)}{\phi(q)}$$ for $q \leq x^{1/3-\epsilon}$ that is a large power of $a$.


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