Suppose the most general form of the Riemann hypothesis were established, say, the Generalized Riemann Hypothesis (GRH), or even the "Grand Riemann Hypothesis." How many of the various unsolved problems concerning prime numbers would be thereby resolved?

I know that the GRH is now known to imply that every odd number $\ge 7$ is the sum of three primes. And other conjectures would be settled with the GRH. But my sense is that many unknowns about prime numbers would remain standing even after GRH is proved. I ask because at one time I naively assumed that the Riemann hypothesis was the key to the primes, but now I am not so sure. Would it at least greatly clarify the distribution of the primes?

I am aware this is not a precise question, but perhaps those knowledgeable could educate us.

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    $\begingroup$ Consequences of GRH spelled out in some detail at mathoverflow.net/questions/17209/… --- any questions beyond what's there? See also en.wikipedia.org/wiki/… $\endgroup$ – Gerry Myerson Jan 27 '13 at 1:25
  • $\begingroup$ Thanks, Gerry, I missed that MO question and the amazing answer by Keith Conrad! $\endgroup$ – Joseph O'Rourke Jan 27 '13 at 1:30
  • $\begingroup$ Maybe you should ask what about primes is not proved by GRH? $\endgroup$ – user59761 Jan 27 '13 at 1:39
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    $\begingroup$ @T97778, plenty --- Goldbach, twin primes, primes of form $n^2+1$, prime between $n$ and $n+2(\log n)^2$. $\endgroup$ – Gerry Myerson Jan 27 '13 at 1:43

It says nothing about large prime gaps, because these can occur in clumps followed by many primes close together, and have no real effect on any averages. Note that the standard conjectures are sort of like this: if $p$ is a prime and $q$ the very next prime, then $$ ? ? \; \; q < p + 3 \log^2 p \; \; ? ? $$ Note that this is true for $p=2,$ and all known $p,$ because of the extra 3 coefficient I put in.

P S. See the latest edition of Unsolved Problems in Number Theory by Richard K. Guy. I met him in San Diego. Not much to talk about, really.

  • $\begingroup$ May we all still be attending JMM when we are 96! :-) $\endgroup$ – Joseph O'Rourke Jan 27 '13 at 1:48

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