Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



The hexagonal versus the square lattice

Authors: Pieter Moree and Herman J.J. te Riele
Journal: Math. Comp. 73 (2004), 451-473
MSC (2000): Primary 11N13, 11Y35, 11Y60
Published electronically: June 11, 2003
MathSciNet review: 2034132
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Schmutz Schaller's conjecture regarding the lengths of the hexagonal versus the lengths of the square lattice is shown to be true. The proof makes use of results from (computational) prime number theory.

Using an identity due to Selberg, it is shown that, in principle, the conjecture can be resolved without using computational prime number theory. By our approach, however, this would require a huge amount of computation.

References [Enhancements On Off] (What's this?)

  • 1. B.C. Berndt, Ramanujan's notebooks. Part IV, Springer-Verlag, New York, 1994. MR 95e:11028
  • 2. B.C. Berndt and R.A. Rankin, Ramanujan: Letters and commentary, AMS, Rhode Island, 1995. MR 97c:01034
  • 3. B.C. Berndt and K. Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary, The Andrews Festschrift (Maratea, 1998), (Eds.) D. Foata, 2001, 39-110. MR 2000i:01027
  • 4. J.H. Conway and N.J.A. Sloane, Sphere packings, lattices and groups, Third edition, Grundlehren der Mathematischen Wissenschaften 290, Springer-Verlag, New York, 1999. MR 2000b:11077
  • 5. D.A. Cox, Primes of the form $x^2 + ny^2$. Fermat, class field theory and complex multiplication, Wiley and Sons, Inc., New York, 1989. MR 90m:11016
  • 6. H. Davenport, Multiplicative number theory, Third revised edition, Springer-Verlag, New York, 2000. MR 2001f:11001
  • 7. S. Finch, Mathematical Constants, Cambridge University Press, to appear (2003); website
  • 8. S. Kühnlein, Partial solution of a conjecture of Schmutz. Arch. Math. (Basel) 67 (1996), 164-172. MR 97h:11069
  • 9. E. Landau, Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zur ihrer additiven Zusammensetzung erforderlichen Quadrate, Archiv der Math. und Physik 13 (1908), 305-312.
  • 10. P. Moree, Chebyshev's bias for composite numbers with restricted prime divisors, arXiv:math.NT/0112100, to appear in Math. Comp.
  • 11. P. Moree, On some claims in Ramanujan's `unpublished' manuscript on the partition and tau functions, to appear in The Ramanujan Journal.
  • 12. P. Moree and J. Cazaran, On a claim of Ramanujan in his first letter to Hardy, Exposition. Math. 17 (1999), 289-311. MR 2001c:11103
  • 13. M.R. Murty and N. Saradha, An asymptotic formula by a method of Selberg, C. R. Math. Rep. Acad. Sci. Canada 15 (1993), 273-277. MR 95d:11118
  • 14. A.G. Postnikov, Introduction to analytic number theory, AMS translations of mathematical monographs 68, AMS, Providence, Rhode Island, 1988. MR 89a:11001
  • 15. O. Ramaré, Sur un théorème de Mertens, Manuscripta Math. 108 (2002), 495-513.
  • 16. O. Ramaré and R. Rumely, Primes in arithmetic progressions, Math. Comp. 65 (1996), 397-425. MR 97a:11144
  • 17. P. Schmutz Schaller, Geometry of Riemann surfaces based on closed geodesics, Bull. Amer. Math. Soc. (N.S.) 35 (1998), 193-214. MR 99b:11098
  • 18. P. Schmutz Schaller, Platonische Körper, Kugelpackungen und hyperbolische Geometrie, Math. Semesterber. 47 (2000), 75-87. MR 2001m:51026
  • 19. A. Selberg, Collected papers, Vol. II, Springer-Verlag, Berlin, 1991. MR 95g:01032
  • 20. J.-P. Serre, Divisibilité de certaines fonctions arithmétiques, Enseignement Math. 22 (1976), 227-260. MR 55:7958
  • 21. D. Shanks, The second-order term in the asymptotic expansion of $B(x)$, Math. Comp. 18 (1964), 75-86. MR 28:2391
  • 22. D. Shanks and L.P. Schmid, Variations on a theorem of Landau. I, Math. Comp. 20 (1966), 551-569. MR 35:1564
  • 23. P. Shiu, Counting sums of two squares: the Meissel-Lehmer method, Math. Comp. 47 (1986), 351-360. MR 87h:11127
  • 24. P. Shiu, Private communication, February 2002.
  • 25. J.M. Song, Sums over nonnegative multiplicative functions over integers without large prime factors. I, Acta Arith. 97 (2001), 329-351. MR 2002f:11130
  • 26. G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, Cambridge, 1995. MR 97e:11005b

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11N13, 11Y35, 11Y60

Retrieve articles in all journals with MSC (2000): 11N13, 11Y35, 11Y60

Additional Information

Pieter Moree
Affiliation: Korteweg–de Vries Institute, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands

Herman J.J. te Riele
Affiliation: CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands

Received by editor(s): May 2, 2002
Received by editor(s) in revised form: August 6, 2002
Published electronically: June 11, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society