The Skewes Number for Twin Primes: Counting Sign Changes of π2(x) – C2Li2(x)
Group of Mathematical Methods in Physics
University of Wrocław
Pl. Maxa Borna 9, PL-50-204 Wrocław, Poland
e-mail: mwolf@ift.uni.wroc.pl
Received:
Received: 05 April 2011; revised: 12 June 2011; accepted: 15 August 2011; published online: 5 October 2011
DOI: 10.12921/cmst.2011.17.01.87-92
OAI: oai:lib.psnc.pl:742
Abstract:
The results of computer investigation of the sign changes of the difference between the number of twin primes π2 (x) and the Hardy-Littlewood conjecture C2Li2 (x) are reported. It turns out that d2 (x) = π2 (x) − C2Li2 (x) changes the sign at unexpectedly low values of x and for x < 248 = 2.81… × 1014 there are 477118 sign changes of this difference. It is conjectured that the number of sign changes of d2 (x) for x ∈(1, T ) is given by T log(T). The running logarithmic densities of the sets for which d2 (x) > 0 and d2 (x) < 0 are plotted for x up to 248.
Key words:
References:
[1] R.F. Arenstorf, There are infinitely many prime twins, 26-th May 2004. http: //arxiv.org/abs/math/0405509v1.
[2] C. Bays, R.H. Hudson, A new bound for the smallest x with π(x) − Li ( x). Mathematics of Computation 69: 1285-1296, (2000). available from http://www.ams. org/mcom/2000-69-23 /S0025-5718-99- 01104-7/S0025-5718-99-01104-7.pdf.
[3] B.C. Berndt, Ramanujan’s Notebooks, Part IV. Springer Verlag (1994).
[4] R.P. Brent, Irregularities in the distribution of primes and twin primes. Mathematics of Computation 29: 43-56 (1975). available from http://wwwmaths.anu.edu.au/ ~brent/p /rpb024.pdf.
[5] A.M. Cohen, M.J.E. Mayhew, On the difference π(x) − Li ( x). Proc. London Math. Soc. 18: 691-713 (1968).
[6] D.W. DeTemple, A quicker convergence to Euler’s constant. The American Mathematical Monthly 100 (5): 468-470 (1993).
[7] W. Ellison, F. Ellison, Prime Numbers. John Wiley and Son (1985).
[8] A. Granville, G. Martin, Prime number races. American Mathematical Monthly 113: 1-33 (2006).
[9] G.H. Hardy, J.E. Littlewood, Some problems of ‘Partitio Numerorum’ III: On the expression of a number as a sum of primes. Acta Mathematica 44: 1-70 (1922).
[10] J. Havil, Gamma: Exploring Euler’s Constant. Princeton University Press, Princeton, NJ (2003).
[11] A.E. Ingham, A note on the distribution of primes. Acta Arithmetica I: 201-211, 1936. available from http://matwbn.icm.edu.pl/ksiazki/aa/aa1/aa1116.pdf.
[12] A.E. Ingham, The distribution of prime numbers. unchanged reprint: Hafner Publ. Comp. (New York) (1971).
[13] J. Kaczorowski, On sign-changes in the remainder-term of the prime-number formula. I. Acta Arithetica 44: 365-377 (1984). available from http://matwbn.icm.edu.pl/ ksiazki/aa/aa44/aa4446.pdf.
[14] J. Kaczorowski, On sign-changes in the remainder-term of the prime-number formula. II. Acta Arithetica 45: 65-74 (1984). available from http://matwbn.icm.edu.pl/ ksiazki/aa/aa45/aa4517.pdf.
[15] J. Kaczorowski, K. Wiertelak, Oscillations of a given size of some arithmetic error terms. Trans. Amer. Math. Soc. 361: 5023-5039 (2009).
[16] S. Knapowski, On sign changes of the difference π(x) −Li ( x). Acta Arithmetica VII: 106-119 (1962).
[17] J. Koreevar, Distributional Wiener-Ikehara theorem and twin primes. Indag. Mathem., N.S. 16: 3749, 2005. Available from http://staff.science.uva.nl/~korevaar/DisWieIke.pdf.
[18] J. Korevaar, H. te Riele. Average prime-pair counting formula. Math. Comput. 79 (270): 1209-1229 (2010).
[19] R.S. Lehman, On the difference π(x) – Li(x). Acta Arithmetica XI: 397-410 (1966).
http://matwbn.icm.edu.pl/ksiazki/aa/aa11/aa11132.pdf.
[20] J.E. Littlewood, Sur la distribution des nombres premieres. Comptes Rendus 158: 1869-1872 (1914).
[21] J. Pintz, On the remainder term of the prime number formula. III. Studia Sci. Math. Hungar. 12: 343-369 (1977).
[22] J. Pintz, On the remainder term of the prime number formula. IV. Studia Sci. Math. Hungar. 13: 29-42 (1978).
[23] H.W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, New York, NY (1986).
[24] P. Ribenboim, The Little Book of Big Primes. 2ed., Springer (2004).
[25] M. Rubinstein, A simple heuristic proof of Hardy and Littlewood conjecture B. Amer. Math. Monthly, 100: 456-460 (1993).
[26] M. Rubinstein, P. Sarnak, Chebyshevs bias. Experimental Mathematics 3: 173-197 (1994).
[27] Y. Saouter, P. Demichel, A sharp region where π(x) − li ( x). is positive. Math. Comput. 79 (272): 2395-2405 (2010).
[28] J.C. Schlage-Puchta, Sign changes of π(x, q,1) − π(x, q, a). Acta Mathematica Hungarica, 102: 305-320 (2004).
[29] S. Skewes, On the difference π(x) − Li ( x). J. London Math. Soc. 8: 277-283, 1934. available from http://www.ift.uni.wroc.pl/~mwolf/Skewes1933.pdf.
[30] S. Skewes. On the difference π(x) − Li ( x). II. Proc. London Math. Soc. 5: 48-70, 1955. available from http://www.ift.uni.wroc.pl/~mwolf/Skewes1955.pdf. 12
[31] H.J. te Riele, On the difference π(x) − Li ( x). Mathematics of Computation 48: 323-328 (1987).
[32] G. Tenenbaum, Re: Arenstorf’s paper on the twin prime conjecture. NM- BRTHRY@listserv.nodak.edu mailing list. 8 Jun 2004. http://listserv.nodak. edu/cgi-bin/wa.exe?
A2=ind0406&L=nmbrthry&F=&S=&P=1119.
[33] P. Turan, On the twin-prime problem II. Acta Arithmetica XIII: 61-89, 1967. available from
http://matwbn.icm.edu.pl/ksiazki/aa/aa13/aa1315.pdf. 13
The results of computer investigation of the sign changes of the difference between the number of twin primes π2 (x) and the Hardy-Littlewood conjecture C2Li2 (x) are reported. It turns out that d2 (x) = π2 (x) − C2Li2 (x) changes the sign at unexpectedly low values of x and for x < 248 = 2.81… × 1014 there are 477118 sign changes of this difference. It is conjectured that the number of sign changes of d2 (x) for x ∈(1, T ) is given by T log(T). The running logarithmic densities of the sets for which d2 (x) > 0 and d2 (x) < 0 are plotted for x up to 248.
Key words:
References:
[1] R.F. Arenstorf, There are infinitely many prime twins, 26-th May 2004. http: //arxiv.org/abs/math/0405509v1.
[2] C. Bays, R.H. Hudson, A new bound for the smallest x with π(x) − Li ( x). Mathematics of Computation 69: 1285-1296, (2000). available from http://www.ams. org/mcom/2000-69-23 /S0025-5718-99- 01104-7/S0025-5718-99-01104-7.pdf.
[3] B.C. Berndt, Ramanujan’s Notebooks, Part IV. Springer Verlag (1994).
[4] R.P. Brent, Irregularities in the distribution of primes and twin primes. Mathematics of Computation 29: 43-56 (1975). available from http://wwwmaths.anu.edu.au/ ~brent/p /rpb024.pdf.
[5] A.M. Cohen, M.J.E. Mayhew, On the difference π(x) − Li ( x). Proc. London Math. Soc. 18: 691-713 (1968).
[6] D.W. DeTemple, A quicker convergence to Euler’s constant. The American Mathematical Monthly 100 (5): 468-470 (1993).
[7] W. Ellison, F. Ellison, Prime Numbers. John Wiley and Son (1985).
[8] A. Granville, G. Martin, Prime number races. American Mathematical Monthly 113: 1-33 (2006).
[9] G.H. Hardy, J.E. Littlewood, Some problems of ‘Partitio Numerorum’ III: On the expression of a number as a sum of primes. Acta Mathematica 44: 1-70 (1922).
[10] J. Havil, Gamma: Exploring Euler’s Constant. Princeton University Press, Princeton, NJ (2003).
[11] A.E. Ingham, A note on the distribution of primes. Acta Arithmetica I: 201-211, 1936. available from http://matwbn.icm.edu.pl/ksiazki/aa/aa1/aa1116.pdf.
[12] A.E. Ingham, The distribution of prime numbers. unchanged reprint: Hafner Publ. Comp. (New York) (1971).
[13] J. Kaczorowski, On sign-changes in the remainder-term of the prime-number formula. I. Acta Arithetica 44: 365-377 (1984). available from http://matwbn.icm.edu.pl/ ksiazki/aa/aa44/aa4446.pdf.
[14] J. Kaczorowski, On sign-changes in the remainder-term of the prime-number formula. II. Acta Arithetica 45: 65-74 (1984). available from http://matwbn.icm.edu.pl/ ksiazki/aa/aa45/aa4517.pdf.
[15] J. Kaczorowski, K. Wiertelak, Oscillations of a given size of some arithmetic error terms. Trans. Amer. Math. Soc. 361: 5023-5039 (2009).
[16] S. Knapowski, On sign changes of the difference π(x) −Li ( x). Acta Arithmetica VII: 106-119 (1962).
[17] J. Koreevar, Distributional Wiener-Ikehara theorem and twin primes. Indag. Mathem., N.S. 16: 3749, 2005. Available from http://staff.science.uva.nl/~korevaar/DisWieIke.pdf.
[18] J. Korevaar, H. te Riele. Average prime-pair counting formula. Math. Comput. 79 (270): 1209-1229 (2010).
[19] R.S. Lehman, On the difference π(x) – Li(x). Acta Arithmetica XI: 397-410 (1966).
http://matwbn.icm.edu.pl/ksiazki/aa/aa11/aa11132.pdf.
[20] J.E. Littlewood, Sur la distribution des nombres premieres. Comptes Rendus 158: 1869-1872 (1914).
[21] J. Pintz, On the remainder term of the prime number formula. III. Studia Sci. Math. Hungar. 12: 343-369 (1977).
[22] J. Pintz, On the remainder term of the prime number formula. IV. Studia Sci. Math. Hungar. 13: 29-42 (1978).
[23] H.W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, New York, NY (1986).
[24] P. Ribenboim, The Little Book of Big Primes. 2ed., Springer (2004).
[25] M. Rubinstein, A simple heuristic proof of Hardy and Littlewood conjecture B. Amer. Math. Monthly, 100: 456-460 (1993).
[26] M. Rubinstein, P. Sarnak, Chebyshevs bias. Experimental Mathematics 3: 173-197 (1994).
[27] Y. Saouter, P. Demichel, A sharp region where π(x) − li ( x). is positive. Math. Comput. 79 (272): 2395-2405 (2010).
[28] J.C. Schlage-Puchta, Sign changes of π(x, q,1) − π(x, q, a). Acta Mathematica Hungarica, 102: 305-320 (2004).
[29] S. Skewes, On the difference π(x) − Li ( x). J. London Math. Soc. 8: 277-283, 1934. available from http://www.ift.uni.wroc.pl/~mwolf/Skewes1933.pdf.
[30] S. Skewes. On the difference π(x) − Li ( x). II. Proc. London Math. Soc. 5: 48-70, 1955. available from http://www.ift.uni.wroc.pl/~mwolf/Skewes1955.pdf. 12
[31] H.J. te Riele, On the difference π(x) − Li ( x). Mathematics of Computation 48: 323-328 (1987).
[32] G. Tenenbaum, Re: Arenstorf’s paper on the twin prime conjecture. NM- BRTHRY@listserv.nodak.edu mailing list. 8 Jun 2004. http://listserv.nodak. edu/cgi-bin/wa.exe?
A2=ind0406&L=nmbrthry&F=&S=&P=1119.
[33] P. Turan, On the twin-prime problem II. Acta Arithmetica XIII: 61-89, 1967. available from
http://matwbn.icm.edu.pl/ksiazki/aa/aa13/aa1315.pdf. 13
