On almost universal mixed sums of squares and triangular numbers

Kane, Ben; Sun, Zhi-Wei

doi:10.1090/S0002-9947-2010-05290-0

On almost universal mixed sums of squares and triangular numbers
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Trans. Amer. Math. Soc. 362 (2010), 6425-6455 Request permission

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

Bruce C. Berndt, Number theory in the spirit of Ramanujan, Student Mathematical Library, vol. 34, American Mathematical Society, Providence, RI, 2006. MR 2246314, DOI 10.1090/stml/034
V. Blomer, G. Harcos, and P. Michel, A Burgess-like subconvex bound for twisted $L$-functions, Forum Math. 19 (2007), no. 1, 61–105. Appendix 2 by Z. Mao. MR 2296066, DOI 10.1515/FORUM.2007.003
L. E. Dickson, Integers represented by positive ternary quadratic forms, Bull. Amer. Math. Soc. 33 (1927), no. 1, 63–70. MR 1561323, DOI 10.1090/S0002-9904-1927-04312-9
L. E. Dickson, History of the Theory of Numbers, Vol. II, AMS Chelsea Publ., 1999.
W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math. 92 (1988), no. 1, 73–90. MR 931205, DOI 10.1007/BF01393993
William Duke and Rainer Schulze-Pillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Invent. Math. 99 (1990), no. 1, 49–57. MR 1029390, DOI 10.1007/BF01234411
A. G. Earnest and J. S. Hsia, Spinor norms of local integral rotations. II, Pacific J. Math. 61 (1975), no. 1, 71–86. MR 404142
A. G. Earnest, J. S. Hsia, and D. C. Hung, Primitive representations by spinor genera of ternary quadratic forms, J. London Math. Soc. (2) 50 (1994), no. 2, 222–230. MR 1291733, DOI 10.1112/jlms/50.2.222
Song Guo, Hao Pan, and Zhi-Wei Sun, Mixed sums of squares and triangular numbers. II, Integers 7 (2007), A56, 5. MR 2373118
Henryk Iwaniec, Fourier coefficients of modular forms of half-integral weight, Invent. Math. 87 (1987), no. 2, 385–401. MR 870736, DOI 10.1007/BF01389423
Burton W. Jones, The Arithmetic Theory of Quadratic Forms, Carcus Monograph Series, no. 10, Mathematical Association of America, Buffalo, N.Y., 1950. MR 0037321
B. Kane, On two conjectures about mixed sums of squares and triangular numbers, J. Combin. Number Theory 1(2009), 77–90.
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.
Martin Kneser, Darstellungsmasse indefiniter quadratischer Formen, Math. Z. 77 (1961), 188–194 (German). MR 140487, DOI 10.1007/BF01180172
Winfried Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), no. 2, 237–268. MR 783554, DOI 10.1007/BF01455989
Byeong-Kweon Oh and Zhi-Wei Sun, Mixed sums of squares and triangular numbers. III, J. Number Theory 129 (2009), no. 4, 964–969. MR 2499416, DOI 10.1016/j.jnt.2008.10.002
O. T. O’Meara, Introduction to Quadratic Forms, Springer, New York, 1963.
Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and $q$-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020489
Ken Ono, Honoring a gift from Kumbakonam, Notices Amer. Math. Soc. 53 (2006), no. 6, 640–651. MR 2235326
Ken Ono and K. Soundararajan, Ramanujan’s ternary quadratic form, Invent. Math. 130 (1997), no. 3, 415–454. MR 1483991, DOI 10.1007/s002220050191
Laurenţiu Panaitopol, On the representation of natural numbers as sums of squares, Amer. Math. Monthly 112 (2005), no. 2, 168–171. MR 2121327, DOI 10.2307/30037415
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.
Rainer Schulze-Pillot, Representation by integral quadratic forms—a survey, Algebraic and arithmetic theory of quadratic forms, Contemp. Math., vol. 344, Amer. Math. Soc., Providence, RI, 2004, pp. 303–321. MR 2060206, DOI 10.1090/conm/344/06226
Rainer Schulze-Pillot, Darstellung durch Spinorgeschlechter ternärer quadratischer Formen, J. Number Theory 12 (1980), no. 4, 529–540 (German, with English summary). MR 599822, DOI 10.1016/0022-314X(80)90043-8
Rainer Schulze-Pillot, Exceptional integers for genera of integral ternary positive definite quadratic forms, Duke Math. J. 102 (2000), no. 2, 351–357. MR 1749442, DOI 10.1215/S0012-7094-00-10227-X
C. L. Siegel, Über die Klassenzahl algebraischer Zahlkörper, Acta Arith. 1(1935), 83–86.
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127 (2007), no. 2, 103–113. MR 2289977, DOI 10.4064/aa127-2-1
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
Z. W. Sun, On sums of primes and triangular numbers, J. Combin. Number Theory 1(2009), 65–76.
J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), no. 4, 375–484 (French). MR 646366