Prime numbers in logarithmic intervals
Authors:
Danilo Bazzanella, Alessandro Languasco and Alessandro Zaccagnini
Journal:
Trans. Amer. Math. Soc. 362 (2010), 26672684
MSC (2010):
Primary 11N05; Secondary 11A41
Published electronically:
November 17, 2009
MathSciNet review:
2584615
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Abstract: Let be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type , where is a prime number and . Then we will apply this to prove that for every there exists a positive proportion of primes such that the interval contains at least a prime number. As a consequence we improve Cheer and Goldston's result on the size of real numbers with the property that there is a positive proportion of integers such that the interval contains no primes. We also prove other results concerning the moments of the gaps between consecutive primes and about the positive proportion of integers such that the interval contains at least a prime number. The last applications of these techniques are two theorems (the first one unconditional and the second one in which we assume the validity of the Riemann Hypothesis and of a form of the Montgomery pair correlation conjecture) on the positive proportion of primes such that the interval contains no primes.
 1.
E.
Bombieri and H.
Davenport, Small differences between prime numbers, Proc. Roy.
Soc. Ser. A 293 (1966), 1–18. MR 0199165
(33 #7314)
 2.
A.
Y. Cheer and D.
A. Goldston, Longer than average intervals
containing no primes, Trans. Amer. Math.
Soc. 304 (1987), no. 2, 469–486. MR 911080
(89d:11073), http://dx.doi.org/10.1090/S00029947198709110807
 3.
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 (88e:11078)
 4.
Jing
Run Chen, On the Goldbach’s problem and the sieve
methods, Sci. Sinica 21 (1978), no. 6,
701–739. MR
517935 (80b:10069)
 5.
P.
Erdös, The difference of consecutive primes, Duke Math.
J. 6 (1940), 438–441. MR 0001759
(1,292h)
 6.
É.
Fouvry and F.
Grupp, On the switching principle in sieve theory, J. Reine
Angew. Math. 370 (1986), 101–126. MR 852513
(87j:11092)
 7.
J.
B. Friedlander and D.
A. Goldston, Some singular series averages and the distribution of
Goldbach numbers in short intervals, Illinois J. Math.
39 (1995), no. 1, 158–180. MR 1299655
(95i:11115)
 8.
P.
X. Gallagher, On the distribution of primes in short
intervals, Mathematika 23 (1976), no. 1,
4–9. MR
0409385 (53 #13140)
 9.
P.
X. Gallagher, Corrigendum: “On the distribution of primes in
short intervals”\ [Mathematika 23 (1976), no. 1, 4–9;\ MR 53
#13140], Mathematika 28 (1981), no. 1, 86. MR 632799
(82j:10072), http://dx.doi.org/10.1112/S0025579300015382
 10.
D.A. Goldston, J. Pintz, and C.Y. Yıldırım.
Primes in Tuples I. to appear in Ann. Math, 2005. http://arxiv.org/abs/math/0508185.
 11.
D.
A. Goldston and C.
Y. Yıldırım, Higher correlations of divisor
sums related to primes. III. Small gaps between primes, Proc. Lond.
Math. Soc. (3) 95 (2007), no. 3, 653–686. MR 2368279
(2008i:11116), http://dx.doi.org/10.1112/plms/pdm021
 12.
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)
 13.
G.H. Hardy and J.E. Littlewood.
Some problems of Partitio Numerorum: VII. Unpublished, 1926.
 14.
D.
R. HeathBrown, Gaps between primes, and the pair correlation of
zeros of the zeta function, Acta Arith. 41 (1982),
no. 1, 85–99. MR 667711
(83m:10078)
 15.
M.
N. Huxley, On the difference between consecutive primes,
Invent. Math. 15 (1972), 164–170. MR 0292774
(45 #1856)
 16.
M.
N. Huxley, Small differences between consecutive primes,
Mathematika 20 (1973), 229–232. MR 0352021
(50 #4509)
 17.
N.
I. Klimov, Combination of elementary and analytic methods in the
theory of numbers, Uspehi Mat. Nauk (N.S.) 13 (1958),
no. 3 (81), 145–164 (Russian). MR 0097372
(20 #3841)
 18.
Helmut
Maier, Small differences between prime numbers, Michigan Math.
J. 35 (1988), no. 3, 323–344. MR 978303
(90e:11126), http://dx.doi.org/10.1307/mmj/1029003814
 19.
A.
Perelli and S.
Salerno, On an average of primes in short intervals, Acta
Arith. 42 (1982/83), no. 1, 91–96. MR 679000
(84b:10064)
 20.
Alberto
Perelli and Saverio
Salerno, On 2𝑘dimensional density estimates, Studia
Sci. Math. Hungar. 20 (1985), no. 14, 345–355.
MR 886039
(88f:11077)
 21.
J.
Barkley Rosser and Lowell
Schoenfeld, Approximate formulas for some functions of prime
numbers, Illinois J. Math. 6 (1962), 64–94. MR 0137689
(25 #1139)
 22.
J.
Wu, Chen’s double sieve, Goldbach’s conjecture and the
twin prime problem, Acta Arith. 114 (2004),
no. 3, 215–273. MR 2071082
(2005e:11128), http://dx.doi.org/10.4064/aa11432
 1.
 E. Bombieri and H. Davenport.
Small differences between prime numbers. Proc. R. Math. Soc., 293:118, 1966. MR 0199165 (33:7314)
 2.
 A.Y. Cheer and D.A. Goldston.
Longer than average intervals containing no primes. Trans. Amer. Math. Soc., 304:469486, 1987. MR 911080 (89d:11073)
 3.
 A.Y. Cheer and D.A. Goldston.
A moment method for primes in short intervals. C.R. Math. Rep. Acad. Sci. Canada, 11:101106, 1987. MR 880600 (88e:11078)
 4.
 J.R. Chen.
On the Goldbach's problem and the sieve methods. Sci. Sinica, 21:701739, 1978. MR 517935 (80b:10069)
 5.
 P. Erdős.
The difference of consecutive primes. Duke Math. J., 6:438441, 1940. MR 0001759 (1:292h)
 6.
 É. Fouvry and F. Grupp.
On the switching principle in sieve theory. J. Reine Angew. Math., 370:101126, 1986. MR 852513 (87j:11092)
 7.
 J.B. Friedlander and D.A. Goldston.
Some singular series averages and the distribution of Goldbach numbers in short intervals. Illinois J. Math., 39:158180, 1995. MR 1299655 (95i:11115)
 8.
 P.X. Gallagher.
On the distribution of primes in short intervals. Mathematika, 23:49, 1976. MR 0409385 (53:13140)
 9.
 P.X. Gallagher.
Corrigendum to: ``On the distribution of primes in short intervals''. Mathematika, 28:86, 1981. MR 632799 (82j:10072)
 10.
 D.A. Goldston, J. Pintz, and C.Y. Yıldırım.
Primes in Tuples I. to appear in Ann. Math, 2005. http://arxiv.org/abs/math/0508185.
 11.
 D.A. Goldston and C.Y. Yıldırım.
Higher correlations of divisor sums related to primes. III. Small gaps between primes. Proc. Lond. Math. Soc. (3), 95(3):653686, 2007. MR 2368279 (2008i:11116)
 12.
 H. Halberstam and H.E. Richert.
Sieve Methods. Academic Press, 1974. MR 0424730 (54:12689)
 13.
 G.H. Hardy and J.E. Littlewood.
Some problems of Partitio Numerorum: VII. Unpublished, 1926.
 14.
 D.R. HeathBrown.
Gaps between primes, and the pair correlation of zeros of the zetafunction. Acta Arith., 41:8599, 1982. MR 667711 (83m:10078)
 15.
 M.N. Huxley.
On the difference between consecutive primes. Invent. Math., 15:155164, 1972. MR 0292774 (45:1856)
 16.
 M.N. Huxley.
Small differences between consecutive primes. Mathematika, 20:229232, 1973. MR 0352021 (50:4509)
 17.
 N.I. Klimov.
Combination of elementary and analytic methods in the theory of numbers. Uspehi Mat. Nauk (N.S.), 13:145164, 1958. (Russian). MR 0097372 (20:3841)
 18.
 H. Maier.
Small differences between prime numbers. Michigan Math. J., 35:323344, 1988. MR 978303 (90e:11126)
 19.
 A. Perelli and S. Salerno.
On an average of primes in short intervals. Acta Arith., 42:9196, 1982. MR 679000 (84b:10064)
 20.
 A. Perelli and S. Salerno.
On dimensional density estimates. Studia Sci. Math. Hungar., 20:345355, 1985. MR 886039 (88f:11077)
 21.
 J.B. Rosser and L. Schoenfeld.
Approximate formulas for some functions of prime numbers. Illinois J. Math., 6:6494, 1962. MR 0137689 (25:1139)
 22.
 J. Wu.
Chen's double sieve, Goldbach's conjecture and the twin prime problem. Acta Arith., 114(3):215273, 2004. MR 2071082 (2005e:11128)
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Additional Information
Danilo Bazzanella
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Email:
danilo.bazzanella@polito.it
Alessandro Languasco
Affiliation:
Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy
Email:
languasco@math.unipd.it
Alessandro Zaccagnini
Affiliation:
Dipartimento di Matematica, Università di Parma, Parco Area delle Scienze, 53/a, Campus Universitario, 43100 Parma, Italy
Email:
alessandro.zaccagnini@unipr.it
DOI:
http://dx.doi.org/10.1090/S0002994709050090
PII:
S 00029947(09)050090
Received by editor(s):
September 17, 2008
Published electronically:
November 17, 2009
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
