Gaps between prime numbers
Authors:
Adolf Hildebrand and Helmut Maier
Journal:
Proc. Amer. Math. Soc. 104 (1988), 19
MSC:
Primary 11N05
MathSciNet review:
958032
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let denote the th gap in the sequence of primes. We show that for every fixed integer and sufficiently large the set of limit points of the sequence in the cube has Lebesgue measure , where is a positive constant depending only on . This generalizes a result of Ricci and answers a question of Erdös, who had asked to prove that the sequence has a finite limit point greater than 1.
 [1]
P. Erdös, On the difference between consecutive primes, Quart. J. Oxford 6 (1935), 124128.
 [2]
P.
Erdös, Problems and results on the differences of consecutive
primes, Publ. Math. Debrecen 1 (1949), 33–37.
MR
0030994 (11,84a)
 [3]
P.
X. Gallagher, A large sieve density estimate near 𝜎=1,
Invent. Math. 11 (1970), 329–339. MR 0279049
(43 #4775)
 [4]
H.
Halberstam and H.E.
Richert, Sieve methods, Academic Press [A subsidiary of
Harcourt Brace Jovanovich, Publishers], LondonNew York, 1974. London
Mathematical Society Monographs, No. 4. MR 0424730
(54 #12689)
 [5]
Helmut
Maier, Chains of large gaps between consecutive primes, Adv.
in Math. 39 (1981), no. 3, 257–269. MR 614163
(82g:10061), http://dx.doi.org/10.1016/00018708(81)900037
 [6]
, Small differences between prime numbers, Preprint.
 [7]
R. A. Rankin, The difference between consecutive prime numbers, J. London Math. Soc. 13 (1938), 242247.
 [8]
Giovanni
Ricci, Recherches sur l’allure de la suite
{𝑝_{𝑛+1}𝑝_{𝑛}/𝑙𝑜𝑔𝑝_{𝑛}},
Colloque sur la Théorie des Nombres, Bruxelles, 1955, Georges
Thone, Liège, 1956, pp. 93–106 (French). MR 0079605
(18,112e)
 [9]
E. Westzynthius, Über die Verteilung der Zahlen, die zu den n ersten Primzahlen teilerfremd sind, Comm. Phys. Math. Helsingfors 25 (1931), 137.
 [1]
 P. Erdös, On the difference between consecutive primes, Quart. J. Oxford 6 (1935), 124128.
 [2]
 , Problems and results on the difference of consecutive primes, Publ. Math. Debrecen 1 (19491950), 3337. MR 0030994 (11:84a)
 [3]
 P. X. Gallagher, A large sieve density estimate near , Invent. Math. 11 (1970), 329339. MR 0279049 (43:4775)
 [4]
 H. Halberstam and H.E. Richert, Sieve methods, Academic Press, New York, 1974. MR 0424730 (54:12689)
 [5]
 H. Maier, Chains of large gaps between consecutive primes, Adv. in Math. 39 (1981), 257269. MR 614163 (82g:10061)
 [6]
 , Small differences between prime numbers, Preprint.
 [7]
 R. A. Rankin, The difference between consecutive prime numbers, J. London Math. Soc. 13 (1938), 242247.
 [8]
 G. Ricci, Recherches sur l'allure de la suite , Colloque sur la Théorie des Nombres, Bruxelles, 1955, pp. 93106. MR 0079605 (18:112e)
 [9]
 E. Westzynthius, Über die Verteilung der Zahlen, die zu den n ersten Primzahlen teilerfremd sind, Comm. Phys. Math. Helsingfors 25 (1931), 137.
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
11N05
Retrieve articles in all journals
with MSC:
11N05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198809580325
PII:
S 00029939(1988)09580325
Article copyright:
© Copyright 1988 American Mathematical Society
