## Acta Mathematica

### Primes in tuples II

#### Abstract

We prove that $\mathop{ \lim \inf}\limits_{n \rightarrow \infty} \frac{p_{n+1}-p_{n}}{\sqrt{\log p_{n}} \left(\log \log p_{n}\right)^{2}}< \infty,$where pn denotes the nth prime. Since on average pn+1pn is asymptotically log n, this shows that we can always find pairs of primes much closer together than the average. We actually prove a more general result concerning the set of values taken on by the differences pp′ between primes which includes the small gap result above.

#### Note

The first author was supported in part by an NSF Grant, the second author by OTKA grants No. K 67676, T 43623, T 49693 and the Balaton program, the third author by TÜBITAK.

#### Article information

Source
Acta Math., Volume 204, Number 1 (2010), 1-47.

Dates
Revised: 21 October 2008
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.acta/1485892443

Digital Object Identifier
doi:10.1007/s11511-010-0044-9

Mathematical Reviews number (MathSciNet)
MR2600432

Zentralblatt MATH identifier
1207.11097

Rights

#### Citation

Goldston, Daniel A.; Pintz, János; Yalçin Yıldırım, Cem. Primes in tuples II. Acta Math. 204 (2010), no. 1, 1--47. doi:10.1007/s11511-010-0044-9. https://projecteuclid.org/euclid.acta/1485892443

#### References

• B ombieri, E. & D avenport, H., Small differences between prime numbers. Proc. Roy. Soc. Ser. A, 293 (1966), 1–18.
• D avenport, H., Multiplicative Number Theory. Graduate Texts in Mathematics, 74. Springer, New York, 2000.
• E lliott, P. D. T. A. & H alberstam, H., A conjecture in prime number theory, in Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), pp. 59–72. Academic Press, London, 1970.
• E rdős, P., The difference of consecutive primes. Duke Math. J., 6 (1940), 438–441.
• G allagher, P. X., On the distribution of primes in short intervals. Mathematika, 23 (1976), 4–9.
• G oldfeld, D. M. & S chinzel, A., On Siegel’s zero. Ann. Sc. Norm. Super. Pisa Cl. Sci., 2 (1975), 571–583.
• G oldston, D. A., P intz, J. & Y ıldırım, C. Y., Primes in tuples I. Ann. of Math., 170 (2009), 819–862.
• H alberstam, H. & R ichert, H. E., Sieve Methods. London Math. Soc. Monogr. Ser., 4. Academic Press, London–New York, 1974.
• H ardy, G. H. & L ittlewood, J. E., Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes. Acta Math., 44 (1923), 1–70.
• H eath-Brown, D. R., Prime twins and Siegel zeros. Proc. Lond. Math. Soc., 47 (1983), 193–224.
• _____ Almost-prime k-tuples. Mathematika, 44 (1997), 245–266.
• I vić, A., The Riemann zeta-function. John Wiley, New York, 1985.
• M aier, H., Small differences between prime numbers. Michigan Math. J., 35 (1988), 323–344.
• M ontgomery, H. L., Topics in Multiplicative Number Theory. Lecture Notes in Mathematics, 227. Springer, Berlin–Heidelberg, 1971.
• P intz, J., Elementary methods in the theory of L-functions. II. On the greatest real zero of a real L-function. Acta Arith., 31 (1976), 273–289.
• _____ Elementary methods in the theory of L-functions. VIII. Real zeros of real L-functions. Acta Arith., 33 (1977), 89–98.
• _____ Very large gaps between consecutive primes. J. Number Theory, 63 (1997), 286–301.
• _____ Approximations to the Goldbach and twin prime problem and gaps between consecutive primes, in Probability and Number Theory (Kanazawa, 2005), Adv. Stud. Pure Math., 49, pp. 323–365. Math. Soc. Japan, Tokyo, 2007.
• de P olignac, A., Six propositions arithmologiques d´eduites du crible d’Ératosthène. Nouv. Ann. Math., 8 (1849), 423–429.
• T itchmarsh, E. C., The Theory of the Riemann Zeta-Function. Oxford University Press, New York, 1986.