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  • $\begingroup$ Thanks! I had thought the $Qx$ term was just from the principal-character, so I had that wrong. Thanks for the $\beta $ observation. I think I'm interested in ``minor arc" $\beta $ (I'll edit my question accordingly) so hopefully that term isn't there. $\endgroup$ Commented 20 hours ago
  • $\begingroup$ @tomos Is the $\sqrt{\log x}$ factor OK with you, i.e. the main thing you are interested in is improving the $Q x \sqrt{\log x}$ term under the minor arc assumption? (Not saying I know how to do this, but if someone does it could help them to know if that would suffice.) $\endgroup$ Commented 19 hours ago
  • $\begingroup$ Ye the $\log $'s are not important for me. What I want is any statement that says ``$\psi _\chi (\beta )$ is $\ll \sqrt x$ on average" (or possibly a worse power). The $Qx$ term would amount to $\ll x/q$ on average, which is not bad. But I want these to hold with the $\beta $ twist. Does that make sense? $\endgroup$ Commented 19 hours ago
  • $\begingroup$ @tomos Oh, sorry, I messed up in copying your notation. I meant to include the $\beta$ in the statement I claimed in the large sieve - the point is that the large sieve only depends on the sum of the norm-squared of the sequence $a_n$ being summed against $\chi$, which is unaffected by multiplication by $e(n \beta)$, so is totally uniform in $\beta$. $\endgroup$ Commented 19 hours ago
  • $\begingroup$ Ah right I see - ye that makes more sense now cheers. In this case I would say "the first moment from second moment argument gives error $...+xQ$ which is worse than the $\ll Q^2\sqrt x$ bound, which comes from Vaughan type deceompositions". So I was hoping that it's possible to do better than first moment from second moment. Maybe it isn't though. $\endgroup$ Commented 19 hours ago