Longer than average intervals containing no primes

Authors:
A. Y. Cheer and D. A. Goldston

Journal:
Trans. Amer. Math. Soc. **304** (1987), 469-486

MSC:
Primary 11N05

DOI:
https://doi.org/10.1090/S0002-9947-1987-0911080-7

MathSciNet review:
911080

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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. Ser. A**293**(1966), 1–18. MR**0199165****[2]**A. Y. Cheer and D. A. Goldston,*A moment method for primes in short intervals*, C. R. Math. Rep. Acad. Sci. Canada**9**(1987), no. 2, 101–106. MR**880600****[3]**Jing Run Chen,*On the Goldbach’s problem and the sieve methods*, Sci. Sinica**21**(1978), no. 6, 701–739. MR**517935****[4]**P. Erdös,*The difference of consecutive primes*, Duke Math. J.**6**(1940), 438–441. MR**0001759****[5]**É. Fouvry and F. Grupp,*On the switching principle in sieve theory*, J. Reine Angew. Math.**370**(1986), 101–126. MR**852513****[6]**P. X. Gallagher,*On the distribution of primes in short intervals*, Mathematika**23**(1976), no. 1, 4–9. MR**0409385**, https://doi.org/10.1112/S0025579300016442**[7]**D. A. Goldston,*The second moment for prime numbers*, Quart. J. Math. Oxford Ser. (2)**35**(1984), no. 138, 153–163. MR**745417**, https://doi.org/10.1093/qmath/35.2.153**[8]**Daniel A. Goldston and Hugh L. Montgomery,*Pair correlation of zeros and primes in short intervals*, Analytic number theory and Diophantine problems (Stillwater, OK, 1984) Progr. Math., vol. 70, Birkhäuser Boston, Boston, MA, 1987, pp. 183–203. MR**1018376****[9]**H. Halberstam and H.-E. Richert,*Sieve methods*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1974. London Mathematical Society Monographs, No. 4. MR**0424730****[10]**D. R. Heath-Brown,*The differences between consecutive primes. III*, J. London Math. Soc. (2)**20**(1979), no. 2, 177–178. MR**551442**, https://doi.org/10.1112/jlms/s2-20.2.177**[11]**D. R. Heath-Brown,*Gaps between primes, and the pair correlation of zeros of the zeta function*, Acta Arith.**41**(1982), no. 1, 85–99. MR**667711****[12]**M. Huxley,*Estimating*, Séminaire de Théorie des Nombres, Anneé 1980-1981, no. 18, 1-10.**[13]**N. N. Lebedev,*Special functions and their applications*, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR**0350075****[14]**K. Prachar,*Bemerkungen über Primzahlen in kurzen Reihen*, Acta Arith.**44**(1984), no. 2, 175–180 (German). MR**774097****[15]**Karl Prachar,*Primzahlverteilung*, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957 (German). MR**0087685****[16]**Atle Selberg,*On the normal density of primes in small intervals, and the difference between consecutive primes*, Arch. Math. Naturvid.**47**(1943), no. 6, 87–105. MR**0012624**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
11N05

Retrieve articles in all journals with MSC: 11N05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1987-0911080-7

Keywords:
Prime numbers,
Poisson distribution

Article copyright:
© Copyright 1987
American Mathematical Society