4

Ole Warnaar and Wadim Zudilin write:

The Riemann Hypothesis does not just “do better” than the Prime Number Theorem
— it is generally believed to be as good as it gets”. That is, we, or far-superior 
extraterrestrial civilisations, will never be able to predict the distribution 
of the primes any better than the Riemann Hypothesis does.

My question concerns the part: it is generally believed to be “as good as it gets”.

Assume the Riemann Hypothesis is proved. What, then, has to be proven to show that “it ain't can get better”. How can this be mathematically formulated?

Sophia Antipolis
  • 390
  • 1
    I think that RH does not imply Legendre's conjecture and if the latter says something more than RH about prime distribution then it may get slightly better. – daniel Apr 01 '18 at 14:04
  • 2
    I have great doubts that this claim can be taken serious. Surely , the riemann hypothesis (better : the GENERALIZED riemann hypothesis) is the key for many questions concerning prime numbers. But that nothing can be found out which cannot be found out with the Riemann hypothesis, well I just can't believe it. – Peter Apr 01 '18 at 14:12
  • 1
    @Peter Both authors are acknowledged number theorists. And they write "it is generally (!) believed". I ask how to formalize this additional belief. – Sophia Antipolis Apr 01 '18 at 14:23
  • I don't know what they mean with "generally believed" , that no serious mathematician has doubts about this claim ? – Peter Apr 01 '18 at 14:29
  • For me, the text is too broad and not specific enough. They probably mean particular aspects of the primes . As written, it rather appears as advertising for the Riemann-hypothesis. – Peter Apr 01 '18 at 14:32
  • @Peter This is how I understand it - at least that there is a very strong bias towards this belief which also includes the extraterrestrials. – Sophia Antipolis Apr 01 '18 at 14:32
  • Rather speculative what extraterrestrials would know about primes , even to assume that they deal with primes. – Peter Apr 01 '18 at 14:35
  • Of course, I don't deny the usefulness of GRH for the study of the prime numbers. – Peter Apr 01 '18 at 14:37
  • OK, so back to humans. I think there is more behind it, the authors emphasize and illustrate this claim. And it does not matter whether it is true or not: the question is certainly legitimate. Only, how to formulate it correctly? – Sophia Antipolis Apr 01 '18 at 14:37
  • Let us continue this discussion in chat. – Peter Apr 01 '18 at 14:38
  • 1
    If there are unproven propositions concerning primes that do not depend on RH it seems that someone could answer this in a factual way, so no need to close as opinion-based. – daniel Apr 01 '18 at 17:36
  • The fact that the word 'believe' appears in the quote seems to tempt some readers to believe that the question is opinion based. That's complete nonsense. It is the request to formulate mathematically the statement that RH is the best achievable result with regard to the distribution of prime numbers. There is no opinion in the question. – Sophia Antipolis Apr 01 '18 at 18:18
  • 2
    I don't think you can "formulate mathematically" the statement in quotes. I think the assertion can be falsified by exhibiting a strong exception to the claim. But the statement is not really a mathematical theorem (or theorem-like claim). – daniel Apr 01 '18 at 18:33
  • I would say that – Simo Ryu Apr 17 '18 at 16:27

0 Answers0