Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

REPRESENTATIONS BY QUATERNARY QUADRATIC FORMS WITH COEFFICIENTS 1, 2, 5 OR 10

  • Alaca, Ayse (School of Mathematics and Statistics Carleton University) ;
  • Altiary, Mada (School of Mathematics and Statistics Carleton University)
  • Received : 2017.12.04
  • Accepted : 2018.06.12
  • Published : 2019.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

We determine explicit formulas for the number of representations of a positive integer n by quaternary quadratic forms with coefficients 1, 2, 5 or 10. We use a modular forms approach.

Keywords

Table 2.1

DBSHCJ_2019_v34n1_27_t0001.png 이미지

References

  1. A. Alaca, S. Alaca, M. F. Lemire, and K. S. Williams, Nineteen quaternary quadratic forms, Acta Arith. 130 (2007), no. 3, 277-310.
  2. A. Alaca, S. Alaca, M. F. Lemire, and K. S. Williams, The number of representations of a positive integer by certain quaternary quadratic forms, Int. J. Number Theory 5 (2009), no. 1, 13-40. https://doi.org/10.1142/S1793042109001943
  3. A. Alaca, S. Alaca, and K. S. Williams, On the quaternary forms $x^2+y^2+z^2+5t^2$, $x^2+y^2+5z^2+5t^2$ and $x^2+5y^2+5z^2+5t^2$, JP J. Algebra Number Theory Appl. 9 (2007), no. 1, 37-53.
  4. A. Alaca and J. Alanazi, Representations by quaternary quadratic forms with coefficients 1, 2, 7 or 14, Integers 16 (2016), Paper No. A55, 16 pp.
  5. B. C. Berndt, Number Theory in the Spirit of Ramanujan, Student Mathematical Library, 34, American Mathematical Society, Providence, RI, 2006.
  6. B. Gordon and D. Sinor, Multiplicative properties of ${\eta}$-products, in Number theory, Madras 1987, 173-200, Lecture Notes in Math., 1395, Springer, Berlin, 1989.
  7. C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, 1829. Reprinted in Gesammelte Werke Vol. 1, Chelsea, New York, 1969, 49-239.
  8. L. J. P. Kilford, Modular Forms: A Classical and Computational Introduction, second edition, Imperial College Press, London, 2015.
  9. H. D. Kloosterman, On the representation of numbers in the form $ax^2+by^2+cz^2+dt^2$, Proc. London Math. Soc. (2) 25 (1926), 143-173. https://doi.org/10.1112/plms/s2-25.1.143
  10. G. Kohler, Eta Products and Theta Series Identities, Springer Monographs in Mathematics, Springer, Heidelberg, 2011.
  11. G. Ligozat, Courbes modulaires de genre 1, Societe Mathematique de France, Paris, 1975.
  12. J. Liouville, Sur les deux formes $x^2+y^2+2(z^2+t^2)$, J. Math. Pures Appl. 5 (1860), 269-272.
  13. J. Liouville, Sur les deux formes $x^2+y^2+z^2+2t^2$ and $x^2+2(y^2+z^2+t^2)$, J. Math. Pures Appl. 6 (1861), 225-230.
  14. J. Liouville, la forme $x^2+y^2+z^2+5t^2$, J. Math. Pures Appl. 9 (1864), 1-12.
  15. J. Liouville, la forme $x^2+5(y^2+z^2+t^2)$, J. Math. Pures Appl. 9 (1864), 17-22.
  16. J. Liouville, la forme $x^2+y^2+5z^2+5t^2$, J. Math. Pures Appl. 10 (1865), 1-8.
  17. G. Lomadse, Uber die Darstellung der Zahlen durch einige quaternare quadratische Formen, Acta Arith. 5 (1959), 125-170. https://doi.org/10.4064/aa-5-2-125-170
  18. K. Ono, The web of modularity: arithmetic of the coefficients of modular forms and q-series, CBMS Regional Conference Series in Mathematics, 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2004.
  19. W. Stein, Modular Forms, a Computational Approach, Graduate Studies in Mathematics, 79, American Mathematical Society, Providence, RI, 2007.
  20. K. S. Williams, On the representations of a positive integer by the forms $x^2+y^2+z^2+2t^2$ and $x^2+2y^2+2z^2+2t^2$, Int. J. Mod. Math. 3 (2008), no. 2, 225-230.
  21. K. S. Williams, Number Theory in the Spirit of Liouville, London Mathematical Society Student Texts, 76, Cambridge University Press, Cambridge, 2011.