On Hölder maps and prime gaps

Haipeng Chen, Jonathan Fraser

Research output: Contribution to journalArticlepeer-review


Let pn denote the nth prime, and consider the function 1/n → 1/pn which maps the reciprocals of the positive integers bijectively to the reciprocals of the primes. We show that Hölder continuity of this function is equivalent to a parametrised family of Cramér type estimates on the gaps between successive primes. Here the parametrisation comes from the Hölder exponent. In particular, we show that Cramér’s conjecture is equivalent to the map 1/n → 1/pn being Lipschitz. On the other hand, we show that the inverse map 1/pn → 1/n is Hölder of all orders but not Lipschitz and this is independent of Cramér’s conjecture.
Original languageEnglish
Pages (from-to)523-532
JournalReal Analysis Exchange
Issue number2
Publication statusPublished - 8 Nov 2021


  • Prime
  • Prime gaps
  • Cramér’s conjecture
  • Hölder maps
  • Lipschitz maps


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