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Let $$\psi _{\chi }(\beta ):=\sum _{n\leq x}\Lambda (n)\chi (n)e(n\beta ).$$ A result like $$\sum _{1<q\leq Q}\sideset {}{^*}\sum _{\chi \text { mod }(q)}|\psi _\chi (0)|\ll Q^2\sqrt x$$ is classical, which is saying that on average $\psi _\chi (0)\ll \sqrt x$ for non-principal characters. (In this statement there may be another error which is less relevant for $Q$ close to $\sqrt x$, and there should maybe be $q/\phi (q)$ factors somewhere; also there should be $\log x$ factors in errors).

I would like to know if a similar result holds for $\psi _\chi (\beta )$. Are there any results? Is it a much harder problem? Assume GRH if you want. (Note that under GRH the individual bound is $\psi _\chi (\beta )\ll x\sqrt \beta $.) Here $\beta $ comes from a minor arc approximation, so you can also think $\beta $ is not too rational if you want.

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  • $\begingroup$ I think it's what I mean - note $\beta $ would be small. (Or am I being particularly silly?). The GRH bound for $\beta =0$ is $\psi _\chi (0)\ll \sqrt x$ so partial summation gives $\psi _\chi (\beta )\ll \sqrt x\Delta $ where $\Delta :=x\beta $. The result I claimed gives instead $\sqrt {x\Delta }$ (Chapter 18.3 of Montgomery-Vaughan II). $\endgroup$ Commented 21 hours ago

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The large sieve gives $$\sum _{1<q\leq Q}\frac{q}{\phi(q) } \sideset {}{^*}\sum _{\chi \text { mod }(q)}|\psi _\chi (\beta)|^2 \leq (Q^2+ x)\sum_{n\leq x} \Lambda(n)^2 \approx (Q^2+ x) x \log x $$

so that

$$\sum _{1<q\leq Q}\frac{q}{\phi(q) } \sideset {}{^*}\sum _{\chi \text { mod }(q)}|\psi _\chi (\beta)| \ll \sqrt{ Q^2 (Q^2+x) x\log x} \approx Q^2 \sqrt{x\log x} + Q x \sqrt{\log x}.$$

This differs from your desired bound of $Q^2 \sqrt{x}$ by the $\sqrt{\log x}$ factor and the additional $Q x \sqrt{\log x}$ term.

Some additional term is necessary if you want an estimate uniform in $\beta$. If $\beta$ is a rational number with denominator $b$ then the $q = b$ terms contribute $\approx x \sqrt{q}$ by writing $e( \beta n)$ so one cannot obtain a bound better than $x \sqrt{Q}$.

I don't know if one should expect the $\sqrt{\log x}$ factor can be removed, nor what the optimal additional term is.

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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

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