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Differences Between Consecutive Primes
Author(s) -
Peck AS
Publication year - 1998
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611598000021
Subject(s) - mathematics , exponent , lemma (botany) , dirichlet distribution , prime number theorem , combinatorics , mean value theorem (divided differences) , discrete mathematics , number theory , distribution (mathematics) , prime number , picard–lindelöf theorem , mathematical analysis , fixed point theorem , ecology , philosophy , linguistics , poaceae , biology , boundary value problem
Let p n be the n th prime. Then this paper is concerned with proving the following result on the distribution of consecutive primes. Theorem. ∑ p n + 1 − p n > x 1 2, x ⩽ p n ⩽ 2 x ( p n + 1 − p n ) \llx 25 36 + ϵ .The exponent of x in this theorem improves on the work of Heath‐Brown who proved (1) with exponent ¾. Under the Riemann hypothesis one can prove (1) with exponent ½. The proof of the theorem starts with the Heath‐Brown–Linnik identity which leads to a formula giving the number of primes in an interval in terms of coefficients of certain Dirichlet series. I then estimate the coefficients by using, among other things, the information which can be gained from Montgomery's mean value theorem and Huxley's version of the Hal' asz lemma. Furthermore, by using familiar sieve arguments I am able to discard some of the coefficients allowing us to gain an improvement over the previous result of Heath‐Brown. 1991 Mathematics Subject Classification : 11N05.