The difference between consecutive primes, III

Proc. London Math. Soc., 83(3), 532--562.] ) ensures us that each of the intervals

On the explicit formula of Riemann-von Mangoldt, II

J. London Math. Soc., 2(28).] for an account on this subject). A theorem of Schoenfeld [Schoenfeld, 1976 †Schoenfeld, L. 1976

Sharper bounds for the Chebyshev Functions

Math. Comp., 30(134), 337--360.] also tells us that the interval

On the order of magnitude of the difference between consecutive prime numbers

Acta Arith., 2, 23--46.] on probabilistic grounds: the interval

On the normal density of primes in small intervals, and the difference between consecutive primes

Archiv Math. Naturv., B.47(6), 82--105.] assuming the Riemann Hypothesis and replacing

Sharper bounds for the Chebyshev Functions

Math. Comp., 30(134), 337--360.] proved the following.

Theorem (1976)

The main ingredient is the explicit formula and a numerical verification of the Riemann hypothesis. From a numerical point of view, the Riemann Hypothesis is known to hold up to a very large height (and larger than in 1976). [Wedeniwski, 2002 †Wedeniwski, S. 2002Letx be a real number larger than2010760 . Then the intervalcontains at least one prime.]x(1−116597),x]

On the Riemann hypothesis

http://www.zetagrid.net/zeta/announcements.html] and the Zeta grid project verified this hypothesis till height

The

http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf] till height

On the zeros of the Riemann zeta-function in the critical strip. IV

Math. Comp., 46(174), 667--681.] who had conducted such a verification in 1986 till height

A sharp region where {

Math. Comp., 79(272), 2395--2405.] casts some doubts on whether all the zeros where checked. Discussions in 2012 with Dave Platt from the university of Bristol led me to believe that the results of [Wedeniwski, 2002 †Wedeniwski, S. 2002

On the Riemann hypothesis

http://www.zetagrid.net/zeta/announcements.html] can be replicated in a very rigorous setting, but that it may be difficult to do so with the results of [Gourdon & Demichel, 2004 †Gourdon, X., & Demichel, P. 2004

The

http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf] with the hardware at our disposal. In [Ramaré & Saouter, 2003 †Ramaré, O., & Saouter, Y. 2003

Short effective intervals containing primes

J. Number Theory, 98, 10--33.], we used the value

Theorem (2002)

If one is interested in somewhat larger value, the paper [Ramaré & Saouter, 2003 †Ramaré, O., & Saouter, Y. 2003Letx be a real number larger than10726905041 . Then the intervalcontains at least one prime.]x(1−128314000),x]

Short effective intervals containing primes

J. Number Theory, 98, 10--33.] also contains the following.

Theorem (2002)

Increasing the lower bound inLetx be a real number larger thanexp(53) . Then the intervalcontains at least one prime.]x(1−1204879661),x]

The

http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf], we can prove that

Theorem (2004, conditional)

Note that all prime gaps have been computed up toLetx be a real number larger thanexp(60) . Then the intervalcontains at least one prime.]x(1−114500755538),x]

New maximal primes gaps and first occurences

Math. Comp., 68(227), 1311--1315.], extending a result of [Young & Potler, 1989 †Young, A., & Potler, J. 1989

First occurence prime gaps

Math. Comp., 52(185), 221--224.]. The proof of these latter results has an asymptotical part, for

Fast Generation of Prime Numbers and Secure Public-Key Cryptographic Parameters

Journal of Cryptology, 8(3), 123--156.]: we only look at families of numbers whose primality can be established with one or two Fermat-like or Pocklington's congruences. This kind of technique has been already used in a quite similar problem in [Deshouillers et al., 1998 †Deshouillers, J.-M., te Riele, H.J.J., & Saouter, Y. 1998

New experimental results concerning the Goldbach conjecture

In: Buhler, J.P. (ed), Algorithmic number theory. Lect. Notes Comput. Sci., no. 1423.]. The generation technique we use relies on a theorem proven in [Brillhart et al., 1975 †Brillhart, J., Lehmer, D.H., & Selfridge, J.L. 1975

New primality crietria and factorizations for

Math. Comp., 29(130), 620--647.] and enables us to generate dense enough families for the upper part of the range to be investigated. For the remaining (smaller) range, we use theorems of [Jaeschke, 1993 †Jaeschke, G. 1993

On strong pseudoprimes to several bases

Math. Comp., 61(204), 915--926.] that yield a fast primality test (for this limited range).

Theorem (2002)

Under the Riemann Hypothesis, the interval]x−85x√logx,x] contains a prime forx≥2 .

Let us recall here that a second line of approach following the original work of \v Ceby\v sev is still under examination though it does not give results as good as analytical means (see [Costa Pereira, 1989 †Costa Pereira, N. 1989

Elementary estimates for the Chebyshev function

Acta Arith., 52, 307--337.] for the latest result). Collecting references: [McCurley, 1984a †McCurley, K.S. 1984a

Explicit estimates for the error term in the prime number theorem for arithmetic progressions

Math. Comp., 42, 265--285.], [McCurley, 1984b †McCurley, K.S. 1984b

Explicit estimates for

Math. Comp., 42, 287--296.], [Kadiri, 2008 †Kadiri, H. 2008

Short effective intervals containing primes in arithmetic progressions and the seven cube problem

Math. Comp., 77(263), 1733--1748.].

Last updated on December 27th, 2012, by Olivier Ramaré