In the answer to this question, engelbret mentions "a conjecture of H. L. Montgomery (not the one on pair correlations, another one), which implies both the GRH and the Elliott-Halberstam conjecture.". My question is : what does this conjecture says, and where is it stated ? I am also interested in any references that would discuss that conjecture.
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3$\begingroup$ I just saw the following question mathoverflow.net/questions/115438/… by you discussing this very conjecture. $\endgroup$– LuciaCommented Sep 2, 2013 at 16:02
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$\begingroup$ yes, I just noticed it now as well, strange... $\endgroup$– Carlo BeenakkerCommented Sep 2, 2013 at 17:49
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2$\begingroup$ Lucia, Carlo, somehow I didn't realize that the conjecture you mention, and that indeed I knew, was implying both GRH and Elliot-Halberstam, hence was the conjecture engelbert was talking about. This was stupid of me. I kind of imagined that there was a deeper conjecture of Montgomery, on the distribution of zero's on the critical line, which would imply and provide an explanation for the you guys mention. Somehow I am not completely convinced by the answer to my question 115438. $\endgroup$– JoëlCommented Nov 6, 2013 at 2:15
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2 Answers
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I think the conjecture you have in mind is that for an arithmetic progression $a\pmod q$ with $(a,q)=1$ and $x\ge q$ $$ \Psi(x;q,a) = \frac{x}{\phi(q)} + O\Big(\frac{x^{\frac 12+\epsilon}}{\sqrt{q}}\Big). $$ This is stronger than GRH which has the error $O(x^{\frac 12+\epsilon})$, and just summing the errors over all $q\le x^{1-\epsilon}$ you get the Elliott-Halberstam conjecture as well.
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discussion: Limitations to the equidistribution of primes (Friedlander and Granville, 1989) and Small gaps between prime numbers (Soundararajan, 2007) (page 16).
the sources to Montgomery are
Topics in multiplicative number theory (Springer, 1971)
Proc. Symp. Pure Math. 28, 307 (1976)
answered Sep 2, 2013 at 13:15