Average prime-pair counting formula

Authors:
Jaap Korevaar and Herman te Riele

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

Published electronically:
September 25, 2009

MathSciNet review:
2600563

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Abstract | References | Similar Articles | Additional Information

Abstract: Taking , let denote the number of prime pairs with . The prime-pair conjecture of Hardy and Littlewood (1923) asserts that with an explicit constant . There seems to be no good conjecture for the remainders that corresponds to Riemann's formula for . However, there is a heuristic approximate formula for averages of the remainders which is supported by numerical results.

**1.**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 Article posted May 26, 2004; withdrawn June 9, 2004.**2.**Paul T. Bateman and Roger A. Horn,*A heuristic asymptotic formula concerning the distribution of prime numbers*, Math. Comp.**16**(1962), 363–367. MR**0148632**, https://doi.org/10.1090/S0025-5718-1962-0148632-7**3.**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****4.**Carter Bays and Richard H. Hudson,*A new bound for the smallest 𝑥 with 𝜋(𝑥)>𝑙𝑖(𝑥)*, Math. Comp.**69**(2000), no. 231, 1285–1296. MR**1752093**, https://doi.org/10.1090/S0025-5718-99-01104-7**5.**E. Bombieri and H. Davenport,*Small differences between prime numbers*, Proc. Roy. Soc. Ser. A**293**(1966), 1–18. MR**0199165****6.**Richard P. Brent,*Irregularities in the distribution of primes and twin primes*, Math. Comp.**29**(1975), 43–56. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. MR**0369287**, https://doi.org/10.1090/S0025-5718-1975-0369287-1**7.**F. J. van de Bult,*Counts of prime pairs*. Private communication including spreadsheet attachment tot10-tot.csv, February 2007.**8.**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****9.**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****10.**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****11.**Andrew Granville and Greg Martin,*Prime number races*, Amer. Math. Monthly**113**(2006), no. 1, 1–33. MR**2202918**, https://doi.org/10.2307/27641834**12.**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**, https://doi.org/10.1007/BF02403921**13.**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****14.**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****15.**Jacob Korevaar,*Distributional Wiener-Ikehara theorem and twin primes*, Indag. Math. (N.S.)**16**(2005), no. 1, 37–49. MR**2138049**, https://doi.org/10.1016/S0019-3577(05)80013-8**16.**J. Korevaar,*Prime pairs and the zeta function*. J. Approx. Theory**158**(2009), 69-96.**17.**Tadej Kotnik,*The prime-counting function and its analytic approximations: 𝜋(𝑥) and its approximations*, Adv. Comput. Math.**29**(2008), no. 1, 55–70. MR**2420864**, https://doi.org/10.1007/s10444-007-9039-2**18.**J. E. Littlewood,*Sur la distribution des nombres premiers*. C. R. Acad. Sci. Paris**158**(1914), 1869-1872.**19.**Hugh L. Montgomery,*Topics in multiplicative number theory*, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, Berlin-New York, 1971. MR**0337847****20.**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****21.**T. R. Nicely,*Enumeration of the twin-prime pairs to*. See the internet, http://www.trnicely.net, September 2008.**22.**Herman J. J. te Riele,*On the sign of the difference 𝜋(𝑥)-𝑙𝑖(𝑥)*, Math. Comp.**48**(1987), no. 177, 323–328. MR**866118**, https://doi.org/10.1090/S0025-5718-1987-0866118-6**23.**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****24.**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****25.**Marek Wolf,*An analog of the Skewes number for twin primes*. Undated attachment to private communication, July 2008.

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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

DOI:
https://doi.org/10.1090/S0025-5718-09-02312-6

Keywords:
Hardy--Littlewood conjecture,
prime-pair functions,
representation by repeated complex integral,
zeta's complex zeros

Received by editor(s):
February 25, 2009

Received by editor(s) in revised form:
June 5, 2009

Published electronically:
September 25, 2009

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.