Average prime-pair counting formula
HTML articles powered by AMS MathViewer
- by Jaap Korevaar and Herman te Riele PDF
- Math. Comp. 79 (2010), 1209-1229 Request permission
Abstract:
Taking $r>0$, let $\pi _{2r}(x)$ denote the number of prime pairs $(p, p+2r)$ with $p\le x$. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that $\pi _{2r}(x)\sim 2C_{2r} \mathrm {li}_2(x)$ with an explicit constant $C_{2r}>0$. There seems to be no good conjecture for the remainders $\omega _{2r}(x)=\pi _{2r}(x)- 2C_{2r} \mathrm {li}_2(x)$ that corresponds to Riemann’s formula for $\pi (x)-\mathrm {li}(x)$. However, there is a heuristic approximate formula for averages of the remainders $\omega _{2r}(x)$ which is supported by numerical results.References
- R.F. Arenstorf, There are infinitely many prime twins. Available on the internet, at http://arxiv.org/abs/math/0405509v1. Article posted May 26, 2004; withdrawn June 9, 2004.
- Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962), 363–367. MR 148632, DOI 10.1090/S0025-5718-1962-0148632-7
- Paul T. Bateman and Roger A. Horn, Primes represented by irreducible polynomials in one variable, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp. 119–132. MR 0176966
- Carter Bays and Richard H. Hudson, A new bound for the smallest $x$ with $\pi (x)>\textrm {li}(x)$, Math. Comp. 69 (2000), no. 231, 1285–1296. MR 1752093, DOI 10.1090/S0025-5718-99-01104-7
- E. Bombieri and H. Davenport, Small differences between prime numbers, Proc. Roy. Soc. London Ser. A 293 (1966), 1–18. MR 199165, DOI 10.1098/rspa.1966.0155
- Richard P. Brent, Irregularities in the distribution of primes and twin primes, Math. Comp. 29 (1975), 43–56. MR 369287, DOI 10.1090/S0025-5718-1975-0369287-1
- F. J. van de Bult, Counts of prime pairs. Private communication including spreadsheet attachment tot10-tot.csv, February 2007.
- Harold Davenport, Multiplicative number theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000. Revised and with a preface by Hugh L. Montgomery. MR 1790423
- H. M. Edwards, Riemann’s zeta function, Dover Publications, Inc., Mineola, NY, 2001. Reprint of the 1974 original [Academic Press, New York; MR0466039 (57 #5922)]. MR 1854455
- 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
- Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly 113 (2006), no. 1, 1–33. MR 2202918, DOI 10.2307/27641834
- G. H. Hardy and J. E. Littlewood, Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes, Acta Math. 44 (1923), no. 1, 1–70. MR 1555183, DOI 10.1007/BF02403921
- A. E. Ingham, The distribution of prime numbers, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. Reprint of the 1932 original; With a foreword by R. C. Vaughan. MR 1074573
- Aleksandar Ivić, The Riemann zeta-function, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1985. The theory of the Riemann zeta-function with applications. MR 792089
- Jacob Korevaar, Distributional Wiener-Ikehara theorem and twin primes, Indag. Math. (N.S.) 16 (2005), no. 1, 37–49. MR 2138049, DOI 10.1016/S0019-3577(05)80013-8
- J. Korevaar, Prime pairs and the zeta function. J. Approx. Theory 158 (2009), 69–96.
- Tadej Kotnik, The prime-counting function and its analytic approximations: $\pi (x)$ and its approximations, Adv. Comput. Math. 29 (2008), no. 1, 55–70. MR 2420864, DOI 10.1007/s10444-007-9039-2
- J. E. Littlewood, Sur la distribution des nombres premiers. C. R. Acad. Sci. Paris 158 (1914), 1869–1872.
- Hugh L. Montgomery, Topics in multiplicative number theory, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, Berlin-New York, 1971. MR 0337847
- H. L. Montgomery, The pair correlation of zeros of the zeta function, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 181–193. MR 0337821
- T. R. Nicely, Enumeration of the twin-prime pairs to $10^{16}$. See the internet, http://www.trnicely.net, September 2008.
- Herman J. J. te Riele, On the sign of the difference $\pi (x)-\textrm {li}(x)$, Math. Comp. 48 (1987), no. 177, 323–328. MR 866118, DOI 10.1090/S0025-5718-1987-0866118-6
- E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550
- E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759
- Marek Wolf, An analog of the Skewes number for twin primes. Undated attachment to private communication, July 2008.
Additional Information
- Jaap Korevaar
- Affiliation: KdV Institute of Mathematics, University of Amsterdam, Science Park 904, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands
- Email: J.Korevaar@uva.nl
- Herman te Riele
- Affiliation: CWI: Centrum Wiskunde en Informatica, Science Park 123, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands
- Email: Herman.te.Riele@cwi.nl
- Received by editor(s): February 25, 2009
- Received by editor(s) in revised form: June 5, 2009
- Published electronically: September 25, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 79 (2010), 1209-1229
- MSC (2000): Primary 11P32; Secondary 65-05
- DOI: https://doi.org/10.1090/S0025-5718-09-02312-6
- MathSciNet review: 2600563