Longer than average intervals containing no primes
Authors:
A. Y. Cheer and D. A. Goldston
Journal:
Trans. Amer. Math. Soc. 304 (1987), 469486
MSC:
Primary 11N05
MathSciNet review:
911080
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Abstract: We present two methods for proving that there is a positive proportion of intervals which contain no primes and are longer than the average distance between consecutive primes. The first method is based on an argument of Erdös which uses a sieve upper bound for prime twins to bound the density function for gaps between primes. The second method uses known results about the first three moments for the distribution of intervals with a given number of primes. Better results are obtained by assuming that the first moments are Poisson. The related problem of longer than average gaps between primes is also considered.
 [1]
E. Bombieri and H. Davenport, Small differences between prime numbers, Proc. Roy. Soc. London Ser. A 293 (1966), 118. MR 0199165 (33:7314)
 [2]
A. Y. Cheer and D. A. Goldston, A moment method for primes in short intervals, C. R. Math. Rep. Acad. Sci. Canada, 11, (1987) No. 2, 101106. MR 880600 (88e:11078)
 [3]
J. R. Chen, On the Goldbach's problem and the sieve methods, Sci. Sinica 21 (1978), 701739. MR 517935 (80b:10069)
 [4]
P. Erdös, The difference of consecutive primes, Duke Math. J. 6 (1940), 438441. MR 0001759 (1:292h)
 [5]
E. Fouvry and F. Grupp, On the switching principle in sieve theory, J. Reine Angew. Math. 370 (1986), 101126. MR 852513 (87j:11092)
 [6]
P. X. Gallagher, On the distribution of primes in short intervals, Mathematica 23 (1976), 49. MR 0409385 (53:13140)
 [7]
D. A. Goldston, The second moment for prime numbers, Quart. J. Math. Oxford (2), 35 (1984), 153163. MR 745417 (85j:11108)
 [8]
D. A. Goldston and H. L. Montgomery, Pair correlation of zeros and primes in short intervals, Analytic Number Theory and Diophantine Problem, Birkhaüser, Boston, Mass., 1987. MR 1018376 (90h:11084)
 [9]
H. Halberstam and H.E. Richert, Sieve methods, Academic Press, London, 1974. MR 0424730 (54:12689)
 [10]
D. R. HeathBrown, The differences between consecutive primes. III, J. London Math. Soc. [2] 20 (1979), 177178. MR 551442 (81f:10055)
 [11]
, Gaps between primes, and the pair correlation of zeros of the zetafunctions, Acta Arith. 41 (1982), 8599. MR 667711 (83m:10078)
 [12]
M. Huxley, Estimating , Séminaire de Théorie des Nombres, Anneé 19801981, no. 18, 110.
 [13]
N. N. Lebedev, Special functions and their applications, Dover, New York, 1972. MR 0350075 (50:2568)
 [14]
K. Prachar, Bemerkungen über Primzahlen in kurzen Reihen, Acta Arith. 44 (1984), 175180. MR 774097 (86e:11080)
 [15]
, Primzahlverteilung, SpringerVerlag, Berlin, 1957. MR 0087685 (19:393b)
 [16]
A. Selberg, On the normal density of primes in small intervals and the difference between consecutive primes, Arch. Math. Naturvid. 47 (1943), No. 6, 87105. MR 0012624 (7:48e)
 [1]
 E. Bombieri and H. Davenport, Small differences between prime numbers, Proc. Roy. Soc. London Ser. A 293 (1966), 118. MR 0199165 (33:7314)
 [2]
 A. Y. Cheer and D. A. Goldston, A moment method for primes in short intervals, C. R. Math. Rep. Acad. Sci. Canada, 11, (1987) No. 2, 101106. MR 880600 (88e:11078)
 [3]
 J. R. Chen, On the Goldbach's problem and the sieve methods, Sci. Sinica 21 (1978), 701739. MR 517935 (80b:10069)
 [4]
 P. Erdös, The difference of consecutive primes, Duke Math. J. 6 (1940), 438441. MR 0001759 (1:292h)
 [5]
 E. Fouvry and F. Grupp, On the switching principle in sieve theory, J. Reine Angew. Math. 370 (1986), 101126. MR 852513 (87j:11092)
 [6]
 P. X. Gallagher, On the distribution of primes in short intervals, Mathematica 23 (1976), 49. MR 0409385 (53:13140)
 [7]
 D. A. Goldston, The second moment for prime numbers, Quart. J. Math. Oxford (2), 35 (1984), 153163. MR 745417 (85j:11108)
 [8]
 D. A. Goldston and H. L. Montgomery, Pair correlation of zeros and primes in short intervals, Analytic Number Theory and Diophantine Problem, Birkhaüser, Boston, Mass., 1987. MR 1018376 (90h:11084)
 [9]
 H. Halberstam and H.E. Richert, Sieve methods, Academic Press, London, 1974. MR 0424730 (54:12689)
 [10]
 D. R. HeathBrown, The differences between consecutive primes. III, J. London Math. Soc. [2] 20 (1979), 177178. MR 551442 (81f:10055)
 [11]
 , Gaps between primes, and the pair correlation of zeros of the zetafunctions, Acta Arith. 41 (1982), 8599. MR 667711 (83m:10078)
 [12]
 M. Huxley, Estimating , Séminaire de Théorie des Nombres, Anneé 19801981, no. 18, 110.
 [13]
 N. N. Lebedev, Special functions and their applications, Dover, New York, 1972. MR 0350075 (50:2568)
 [14]
 K. Prachar, Bemerkungen über Primzahlen in kurzen Reihen, Acta Arith. 44 (1984), 175180. MR 774097 (86e:11080)
 [15]
 , Primzahlverteilung, SpringerVerlag, Berlin, 1957. MR 0087685 (19:393b)
 [16]
 A. Selberg, On the normal density of primes in small intervals and the difference between consecutive primes, Arch. Math. Naturvid. 47 (1943), No. 6, 87105. MR 0012624 (7:48e)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198709110807
PII:
S 00029947(1987)09110807
Keywords:
Prime numbers,
Poisson distribution
Article copyright:
© Copyright 1987
American Mathematical Society
