Prime numbers in logarithmic intervals

Authors:
Danilo Bazzanella, Alessandro Languasco and Alessandro Zaccagnini

Journal:
Trans. Amer. Math. Soc. **362** (2010), 2667-2684

MSC (2010):
Primary 11N05; Secondary 11A41

DOI:
https://doi.org/10.1090/S0002-9947-09-05009-0

Published electronically:
November 17, 2009

MathSciNet review:
2584615

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Abstract | References | Similar Articles | Additional Information

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. London Ser. A**293**(1966), 1–18. MR**199165**, https://doi.org/10.1098/rspa.1966.0155**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**, https://doi.org/10.1090/S0002-9947-1987-0911080-7**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****4.**Jing Run Chen,*On the Goldbach’s problem and the sieve methods*, Sci. Sinica**21**(1978), no. 6, 701–739. MR**517935****5.**P. Erdös,*The difference of consecutive primes*, Duke Math. J.**6**(1940), 438–441. MR**1759****6.**É. Fouvry and F. Grupp,*On the switching principle in sieve theory*, J. Reine Angew. Math.**370**(1986), 101–126. MR**852513****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****8.**P. X. Gallagher,*On the distribution of primes in short intervals*, Mathematika**23**(1976), no. 1, 4–9. MR**409385**, https://doi.org/10.1112/S0025579300016442**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**, https://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**, https://doi.org/10.1112/plms/pdm021**12.**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****13.**G.H. Hardy and J.E. Littlewood.

Some problems of Partitio Numerorum: VII.*Unpublished*, 1926.**14.**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**, https://doi.org/10.4064/aa-41-1-85-99**15.**M. N. Huxley,*On the difference between consecutive primes*, Invent. Math.**15**(1972), 164–170. MR**292774**, https://doi.org/10.1007/BF01418933**16.**M. N. Huxley,*Small differences between consecutive primes*, Mathematika**20**(1973), 229–232. MR**352021**, https://doi.org/10.1112/S0025579300004836**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****18.**Helmut Maier,*Small differences between prime numbers*, Michigan Math. J.**35**(1988), no. 3, 323–344. MR**978303**, https://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**, https://doi.org/10.4064/aa-42-1-91-96**20.**Alberto Perelli and Saverio Salerno,*On 2𝑘-dimensional density estimates*, Studia Sci. Math. Hungar.**20**(1985), no. 1-4, 345–355. MR**886039****21.**J. Barkley Rosser and Lowell Schoenfeld,*Approximate formulas for some functions of prime numbers*, Illinois J. Math.**6**(1962), 64–94. MR**0137689****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**, https://doi.org/10.4064/aa114-3-2

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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:
https://doi.org/10.1090/S0002-9947-09-05009-0

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.