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Questions here https://mathoverflow.net/questions/198933/zeta-functions-versus-cramers-conjecture, here Cramer and Riemann Conjecture Implication query on relation of Cramer and Riemann zeta functions.

Is there an important consequence that could happen if Cramer's conjecture fails?

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Turbo
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I think I will skip the part about being important. There is a formulation by Nicolas of a simple condition that is equivalent to RH and involves primorials, so it is easier to compute with than Robin's Criterion; Robin was his student.

There is a real number calculated in the criterion of Nicolas that is, as far as anyone has computed, strictly increasing as we got through the primorial numbers. Later, however, Planat et al showed that strict increasing forever would disprove Cramer's conjecture. Meanwhile item 84 in NICOLAS

PLANAT

Let's see, taking consecutive primorials $P,$ the function that, so far, keeps growing is $$ \frac{e^\gamma \; \log \log P \; \; \phi(P)}{P}. $$ The really nice part is that we don't need to calculate $P$ itself. For each new prime factor $p,$ add $\log p$ to update $\log P,$ and multiply by $$ \left( 1 - \frac{1}{p} \right) $$ to update $$ \frac{\phi(P)}{P}. $$ By the time $p=211$ we have Nicolas' ratio reaching $0.97.$

Will Jagy
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  • very interesting. – Turbo Mar 06 '15 at 02:12
  • "A natural approach to derive this inequality would be to prove that a specific sequence related to that bound is strictly decreasing. We show that, unfortunately, this latter fact would contradict Cram´er conjecture on gaps between consecutive primes." Seems to imply RH iumplies no Cramer's conjecture. – Turbo Mar 06 '15 at 02:20
  • @Turbo, not at all. It says that if the sequence of reals (which has lim sup equal to $1$) increases forever, then it is always below $1$ and RH is true, but Cramer is false. If the sequence eventually oscillates, then we have said nothing about Cramer or about RH. In human terms, all this says is that no-one is ever going to prove either possiblity for the sequence of reals. – Will Jagy Mar 06 '15 at 02:30
  • "In human terms, all this says is that no-one is ever going to prove either possiblity for the sequence of reals" seems very pessimistic of human capability. Afterall it just took 2000 years to construct some polygons using straight edge/compass, nowdays even polyhedra building is routine on a 3D machine. – Turbo Mar 06 '15 at 02:45
  • Could you please comment on how you obtained update formula? – Turbo Mar 06 '15 at 07:59
  • @WillJagy So are you biased towards Cramer's conjecture to be false? – Turbo Aug 01 '15 at 13:59
  • @WillJagy It seems that this has complexity theory implication http://cstheory.stackexchange.com/questions/4769/an-np-complete-variant-of-factoring. – Turbo Aug 01 '15 at 14:19
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    @Turbo, my guess is that the sequence eventually oscillates, so there is no resulting comment on either Cramer or RH. – Will Jagy Aug 01 '15 at 18:28
  • http://vixra.org/pdf/1512.0006v1.pdf opinion? – Turbo Dec 06 '15 at 10:26