30 1. Arithmetic Functions H L. Montgomery and R. C. Vaughan [MV07], where the possibility that θ = ε with ε 0 arbitrary is discussed. Also see the paper [Gon93] by S. M. Gonek. An even stronger conclusion follows if one accepts a probabilistic model of the distribu- tion of the primes originated by H. Cram´ er [Cra36], and modified by A. Granville [Gra95] after work of H. Maier [Mai85]. This indicates that there should be some constant C 2e−γ such that the interval (x − C log2(x),x] contains a prime for all x suﬃciently large, and that possibly any C 2e−γ will do. Defining dk = pk+1 − pk one may, in view of the above considerations, ask how large dk can become, say in terms of pk. It is an immediate consequence of the Prime Number Theorem that lim supk→+∞ dk/ log(pk) ≥ c with c = 1. R. J. Backlund [Bac29] showed in 1929 that one can take c = 2, and A. T. Brauer and H. Zeitz [BZ30] achieved c = 4 the year after. E. Westzynthius [Wes31] proved in 1933 that lim sup k→+∞ dk log(p k ) log 3 (p k ) log 4 (pk) ≥ 2eγ, where logm(x) denotes the m times iterated logarithm. This was improved to lim sup k→+∞ dk log(pk) log 3 (pk) 0 by G. Ricci [Ric34]. Further progress was made by Erd¨ os [Erd35], who showed that lim sup k→+∞ dk log(pk) log 2 (pk) (log 3 (pk))2 0, and by Rankin [Ran38], who showed in 1938 that lim sup k→+∞ dk log(pk) log 2 (pk) log 4 (pk) (log 3 (pk))2 ≥ c with c = 1/3. Since then, only improvements in the constant c have been obtained, by A. Sch¨ onhage [Sch63], (c = eγ0/2 in 1963), Rankin [Ran63], (c = eγ0 in 1963), H. Maier and C. B. Pomerance [MP90], (c = 1.312...eγ0 in 1990), and Pintz [Pin97], (c = 2eγ0 in 1997.) The question of how small dk can be in the long run is also of interest. If there are infinitely many twin primes, then dk = 2 infinitely often. Defining E = lim infn→∞ dk/ log(pk), it is again an immediate consequence of the Prime Number Theorem that E ≤ 1. Upper bounds for E were obtained by Erd¨ os [Erd40] (E 1 − c for some small computable c 0 in 1940), Rankin [Ran50] (E ≤ 42/43 in 1950), Ricci [Ric54a] (E ≤ 15/16 in 1954), E. Bombieri and H. Davenport [BD66] (E ≤ (2+ √ 3)/8 in 1966), G. Z. Pil’tja˘ [Pil72] (E ≤ (2 √ 2 − 1)/4 in 1972), Huxley [Hux73, Hux77, Hux84] (E ≤ 1/4 + π/16 in 1973, E ≤ 0.4425 . . . in 1977 and E ≤ 0.4393 . . . in 1984), ´ Fouvry and F. Grupp [FG86] (E ≤ 0.4342 . . . in 1986), H. Maier [Mai88] (E ≤ 0.2484 . . . in 1988) and finally D. A. Goldston, J. Pintz and C. Y. Yıldırım [GPY09] (E = 0 in 2005.) Recently Y. Zhang [Zha14] proved that dk ≤ 70·106 infinitely often, which was a great advance. In work to appear in Annals of Mathematics, J. Maynard has shown that pk+m − pk ≤ cm infinitely often for each positive integer m, with c1 = 600 admissible.

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