Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On almost universal mixed sums of squares and triangular numbers


Authors: Ben Kane and Zhi-Wei Sun
Journal: Trans. Amer. Math. Soc. 362 (2010), 6425-6455
MSC (2010): Primary 11E25; Secondary 11D85, 11E20, 11E95, 11F27, 11F37, 11P99, 11S99
DOI: https://doi.org/10.1090/S0002-9947-2010-05290-0
Published electronically: July 23, 2010
MathSciNet review: 2678981
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than $ 2719$ can be represented by the famous Ramanujan form $ x^2+y^2+10z^2$; equivalently the form $ 2x^2+5y^2+4T_z$ represents all integers greater than 1359, where $ T_z$ denotes the triangular number $ z(z+1)/2$. Given positive integers $ a,b,c$ we employ modular forms and the theory of quadratic forms to determine completely when the general form $ ax^2+by^2+cT_z$ represents sufficiently large integers and to establish similar results for the forms $ ax^2+bT_y+cT_z$ and $ aT_x+bT_y+cT_z$. Here are some consequences of our main theorems: (i) All sufficiently large odd numbers have the form $ 2ax^2+y^2+z^2$ if and only if all prime divisors of $ a$ are congruent to 1 modulo 4. (ii) The form $ ax^2+y^2+T_z$ is almost universal (i.e., it represents sufficiently large integers) if and only if each odd prime divisor of $ a$ is congruent to 1 or 3 modulo 8. (iii) $ ax^2+T_y+T_z$ is almost universal if and only if all odd prime divisors of $ a$ are congruent to 1 modulo 4. (iv) When $ v_2(a)\not=3$, the form $ aT_x+T_y+T_z$ is almost universal if and only if all odd prime divisors of $ a$ are congruent to 1 modulo 4 and $ v_2(a)\not=5,7,\ldots$, where $ v_2(a)$ is the $ 2$-adic order of $ a$.


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

  • 1. B. C. Berndt, Number Theory in the Spirit of Ramanujan, Amer. Math. Soc., Providence, R.I., 2006. MR 2246314 (2007f:11001)
  • 2. V. Blomer, G. Harcos, P. Michel, A Burgess-like subconvex bound for twisted $ L$-functions (with Appendix 2 by Z. Mao), Forum Math. 19(2007), 61-105. MR 2296066 (2008i:11067)
  • 3. L. E. Dickson, Integers represented by positive ternary quadratic forms, Bull. Amer. Math. Soc. 33(1927), 63-70. MR 1561323
  • 4. L. E. Dickson, History of the Theory of Numbers, Vol. II, AMS Chelsea Publ., 1999.
  • 5. W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math. 92(1988), 73-90. MR 931205 (89d:11033)
  • 6. W. Duke and R. Schulze-Pillot, Representations of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Invent. Math. 99(1990), 49-57. MR 1029390 (90m:11051)
  • 7. A. Earnest, J. S. Hsia, Spinor norms of local integral rotations, II, Pacific J. Math. 61(1975), 71-86. MR 0404142 (53:7946)
  • 8. A. Earnest, J. S. Hsia and D. Hung, Primitive representations by spinor genera of ternary quadratic forms, J. London Math. Soc. (2)50(1994), 222-230. MR 1291733 (95k:11044)
  • 9. S. Guo, H. Pan and Z. W. Sun, Mixed sums of squares and triangular numbers (II), Integers 7(2007), # A56, 5pp. MR 2373118 (2008m:11078)
  • 10. H. Iwaniec, Fourier coefficients of modular forms of half-integral weight, Invent. Math. 87(1987), 385-401. MR 870736 (88b:11024)
  • 11. B. Jones, The Arithmetic Theory of Quadratic Forms, Math. Assoc. Amer., Carus Math. Mono. 10, Buffalo, New York, 1950. MR 0037321 (12:244a)
  • 12. B. Kane, On two conjectures about mixed sums of squares and triangular numbers, J. Combin. Number Theory 1(2009), 77-90.
  • 13. H. D. Kloosterman, On the representation of numbers in the form $ ax^2+by^2+cz^2+dt^2$, Acta Math. 49(1926), 407-464.
  • 14. M. Kneser, Darstellungsmasse indefiniter quadratischer Formen, Math. Z. 11(1961), 188-194. MR 0140487 (25:3907)
  • 15. W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271(1985), 237-268. MR 783554 (86i:11018)
  • 16. B. K. Oh and Z. W. Sun, Mixed sums of squares and triangular numbers (III), J. Number Theory 129(2009), 964-969. MR 2499416
  • 17. O. T. O'Meara, Introduction to Quadratic Forms, Springer, New York, 1963.
  • 18. K. Ono, Web of Modularity: Arithmetic of the Coefficients of Modular Forms and $ Q$-series, Amer. Math. Soc., Providence, R.I., 2003. MR 2020489 (2005c:11053)
  • 19. K. Ono, Honoring a gift from Kumbakonam, Notices Amer. Math. Soc. 53(2006), 640-651. MR 2235326
  • 20. K. Ono and K. Soundararajan, Ramanujan's ternary quadratic form, Invent. Math. 130(1997), 415-454. MR 1483991 (99b:11036)
  • 21. L. Panaitopol, On the representation of natural numbers as sums of squares, Amer. Math. Monthly 112(2005), 168-171. MR 2121327 (2005k:11077)
  • 22. S. Ramanujan, On the expression of a number in the form $ ax^2+by^2+cz^2+du^2$, Proc. Camb. Philo. Soc. 19(1916), 11-21.
  • 23. R. Schulze-Pillot, Representations by integral quadratic forms - A Survey, in: Algebraic and Arithmetic Theory of Quadratic Forms, pp. 303-321, Contemp. Math., 344, Amer. Math. Soc., Providence, R.I., 2004. MR 2060206 (2005g:11057)
  • 24. R. Schulze-Pillot, Darstellung durch Spinorgeschlechter ternarer quadratischer Formen, J. Number Theory 12(1980), 529-540. MR 599822 (82k:10024)
  • 25. R. Schulze-Pillot, Exceptional integers for genera of integral ternary positive definite quadratic forms, Duke Math. J. 102(2000), 351-357. MR 1749442 (2001a:11068)
  • 26. C. L. Siegel, Über die Klassenzahl algebraischer Zahlkörper, Acta Arith. 1(1935), 83-86.
  • 27. Z. W. Sun, Mixed sums of squares and triangular numbers, Acta. Arith. 127(2007), 103-113. MR 2289977 (2007m:11052)
  • 28. Z. W. Sun, A message to Number Theory Mailing List, April 27, 2008. http://listserv. nodak.edu/cgi-bin/wa.exe?A2=ind0804&L=nmbrthry&T=0&P=1670
  • 29. Z. W. Sun, On sums of primes and triangular numbers, J. Combin. Number Theory 1(2009), 65-76.
  • 30. J. L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. 60(1981), 375-484. MR 646366 (83h:10061)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11E25, 11D85, 11E20, 11E95, 11F27, 11F37, 11P99, 11S99

Retrieve articles in all journals with MSC (2010): 11E25, 11D85, 11E20, 11E95, 11F27, 11F37, 11P99, 11S99


Additional Information

Ben Kane
Affiliation: Department of Mathematics, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
Email: bkane@math.uni-koeln.de

Zhi-Wei Sun
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email: zwsun@nju.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-2010-05290-0
Keywords: Representations of integers, triangular numbers, sums of squares, quadratic forms, half-integral weight modular forms
Received by editor(s): September 18, 2008
Published electronically: July 23, 2010
Additional Notes: This research was conducted when the first author was a postdoctor at Radboud Universiteit, Nijmegen, Netherlands.
The second author was the corresponding author and he was supported by the National Natural Science Foundation (grant 10871087) of the People’s Republic of China
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society