Unusually large gaps between consecutive primes
Authors:
Helmut Maier and Carl Pomerance
Journal:
Trans. Amer. Math. Soc. 322 (1990), 201237
MSC:
Primary 11N05; Secondary 11N35
MathSciNet review:
972703
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let denote the largest gap between consecutive primes below . In a series of papers from 1935 to 1963, Erdàs, Rankin, and Schànhage showed that , where and is Euler's constant. Here, this result is shown with where is the solution of the equation . The principal new tool used is a result of independent interest, namely, a mean value theorem for generalized twin primes lying in a residue class with a large modulus.
 [1]
N.
G. de Bruijn, On the number of positive integers ≤𝑥 and
free of prime factors >𝑦, Nederl. Acad. Wetensch. Proc.
Ser. A. 54 (1951), 50–60. MR 0046375
(13,724e)
 [2]
H. Cramér, On the order of magnitude of the difference between consecutive prime numbers, Acta Arith. 2 (1936), 396403.
 [3]
P. Erdàs, On the difference of consecutive primes, Quart. J. Math. Oxford Ser. 6 (1935), 124128.
 [4]
P.
X. Gallagher, A large sieve density estimate near 𝜎=1,
Invent. Math. 11 (1970), 329–339. MR 0279049
(43 #4775)
 [5]
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)
 [6]
Henryk
Iwaniec, On the problem of Jacobsthal, Demonstratio Math.
11 (1978), no. 1, 225–231. MR 499895
(80h:10053)
 [7]
A.
F. Lavrik, The number of 𝑘twin primes lying on an interval
of a given length., Soviet Math. Dokl. 2 (1961),
52–55. MR
0132055 (24 #A1902)
 [8]
S.t. Lou and Q. Yao, On gaps between consecutive primes (to appear).
 [9]
Hugh
L. Montgomery, Topics in multiplicative number theory, Lecture
Notes in Mathematics, Vol. 227, SpringerVerlag, BerlinNew York, 1971. MR 0337847
(49 #2616)
 [10]
H.
L. Montgomery and R.
C. Vaughan, The exceptional set in Goldbach’s problem,
Acta Arith. 27 (1975), 353–370. Collection of
articles in memory of Juriĭ\ Vladimirovič Linnik. MR 0374063
(51 #10263)
 [11]
Karl
Prachar, Primzahlverteilung, SpringerVerlag,
BerlinGöttingenHeidelberg, 1957 (German). MR 0087685
(19,393b)
 [12]
R. A. Rankin, The difference between consecutive prime numbers, J. London Math. Soc. 13 (1938), 242247.
 [13]
R.
A. Rankin, The difference between consecutive prime numbers.
V, Proc. Edinburgh Math. Soc. (2) 13 (1962/1963),
331–332. MR 0160767
(28 #3978)
 [14]
Arnold
Schönhage, Eine Bemerkung zur Konstruktion grosser
Primzahllücken, Arch. Math. (Basel) 14 (1963),
29–30 (German). MR 0146154
(26 #3680)
 [15]
Daniel
Shanks, On maximal gaps between successive
primes, Math. Comp. 18 (1964), 646–651. MR 0167472
(29 #4745), http://dx.doi.org/10.1090/S00255718196401674728
 [1]
 N. G. deBruijn, On the number of positive integers and free of prime factors , Nederl. Akad. Wetensch. Proc. Ser. A 54 (1951), 5060. MR 0046375 (13:724e)
 [2]
 H. Cramér, On the order of magnitude of the difference between consecutive prime numbers, Acta Arith. 2 (1936), 396403.
 [3]
 P. Erdàs, On the difference of consecutive primes, Quart. J. Math. Oxford Ser. 6 (1935), 124128.
 [4]
 P. X. Gallagher, A large sieve density estimate near , Invent. Math. 11 (1970), 329339. MR 0279049 (43:4775)
 [5]
 H. Halberstam and H.E. Richert, Sieve methods, Academic Press, London, 1974. MR 0424730 (54:12689)
 [6]
 H. Iwaniec, On the problem of Jacobsthal, Demonstratio Math. 11 (1978), 225231. MR 499895 (80h:10053)
 [7]
 A. F. Lavrik, The number of twin primes lying on an interval of a given length, Dokl. Akad. Nauk SSSR 136 (1961), 281283; English transl., Soviet Math. Dokl. 2 (1961), 5255. MR 0132055 (24:A1902)
 [8]
 S.t. Lou and Q. Yao, On gaps between consecutive primes (to appear).
 [9]
 H. L. Montgomery, Topics in multiplicative number theory, Lecture Notes in Math., vol. 227, SpringerVerlag, Berlin and New York, 1971. MR 0337847 (49:2616)
 [10]
 H. L. Montgomery and R. C. Vaughan, The exceptional set in Goldbach's problem, Acta Arith. 27 (1975), 353370. MR 0374063 (51:10263)
 [11]
 K. Prachar, Primzahlverteilung, SpringerVerlag, Berlin, Gàttingen, and Heidelberg, 1957. MR 0087685 (19:393b)
 [12]
 R. A. Rankin, The difference between consecutive prime numbers, J. London Math. Soc. 13 (1938), 242247.
 [13]
 , The difference between consecutive prime numbers. V, Proc. Edinburgh Math. Soc. (2) 13 (1962/63), 331332. MR 0160767 (28:3978)
 [14]
 A. Schànhage, Eine Bemerkung zur Konstruktion grosser Primzahllücken, Arch. Math. 14 (1963), 2930. MR 0146154 (26:3680)
 [15]
 D. Shanks, On maximal gaps between successive primes, Math. Comp. 18 (1964), 646651. MR 0167472 (29:4745)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
11N05,
11N35
Retrieve articles in all journals
with MSC:
11N05,
11N35
Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719900972703X
PII:
S 00029947(1990)0972703X
Article copyright:
© Copyright 1990
American Mathematical Society
