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I must profess a general ignorance of problems that were once known to be true under GRH but has since became unconditional due to the Bombieri-Vinogradov theorem, but I am aware of the heuristic that Bombieri-Vinogradov (which is sort of a "GRH on average" type result) can be substituted for GRH in some cases.

So my question is... what are some types of problems where the Bombieri-Vinogradov theorem may be used as a substitute for GRH?

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  • $\begingroup$ As you say, B--V is often called "GRH on average". The reason is this: First, think of GRH as a statement about the error term for primes in progressions. B--V says (roughly) that a certain sum of error terms is just as small as it would be if each individual term obeys the GRH-conditional upper bound. In many problems in analytic number theory, the error term at the end of the day is bounded by a sum of error terms for the PNT in AP. For example, this is the case in a lot of sieve problems. Often, such a sum is close enough to the sum estimated by B--V that one can cite it directly. $\endgroup$ Jan 9, 2014 at 3:11

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