The quantitative behaviour of polynomial orbits on nilmanifolds « What's new - The Diigo Meta page

• 15 Jul 08
  

  arithwsun arithwsun

  For instance, the question of whether (10^n \pi)_{n \in {\Bbb N}} is equidistributed mod 1 is an old unsolved problem, equivalent to asking whether \pi is normal base 10.

  twin-primes uniform-distribution nilmanifolds nilsequences expository remark featured

  • For instance, the question of whether is equidistributed mod 1 is an old unsolved problem, equivalent to asking whether is normal base 10.
  • For instance, the question of whether is equidistributed mod 1 is an old unsolved problem, equivalent to asking whether is normal base 10.

  • The difficulty of the twin-prime problem compared to the current best results in additive prime number theory may be somewhat suggested by the fact that the best result of the type

     

    (Known widely believed conjecture) implies (Twin Prime-like Conjecture)

     

    which is currently available is the consequence of the work of Goldston-Pintz-Yildirim according to which the Elliott-Halberstam conjecture implies that for some (small, less than 16 I think currently) fixed even integer $k$, there are infinitely many primes $p$ for which $p+k$ is also prime. The E-H conjecture is (at least in appearance) much stronger than what can be derived from the Generalized Riemann Hypothesis for all Dirichlet L-functions. Moreover, this conjecture is clearly borderline; more optimistic statements that might seem just as reasonable are known to be false by work of Friedlander and Granville, suggested by earlier works of Maier on primes in short intervals.
  • [Incidentally, regarding the interactions between physics and number theory: physical intuition has proven to be quite useful in making accurate predictions about many mathematical objects, such as the distribution of zeroes of the Riemann zeta function, but has been significantly less useful in generating rigorous proofs of these predictions. In number theory, our ability to make accurate predictions on anything relating to the primes (or related objects) is now remarkably good, but our ability to actually prove these predictions rigorously lags behind quite significantly. So I doubt that the key to further rigorous progress on these problems lies with physics.]

  2 more annotations...

Would you like to comment?

Join Diigo for a free account, or sign in if you are already a member.