Abstract
Let f(x1, . . . , xs) (s ≥ 3) be a regular quadratic form with integral variables. We study the number of divisors of f(x1, . . . , xs) on average. We establish an asymptotic formula of the sum of these divisors.
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B. J. Birch, Forms in many variables, Proc. R. Soc. Lond., Ser. A, 265(1321):245–263, 1961.
C. Calderón and M. J. de Velasco, On divisors of a quadratic form, Bol. Soc. Bras. Mat., 31(1):81–91, 2000.
F. Chamizo and H. Iwaniec, On the sphere problem, Rev. Mat. Iberoamer., 11(2):417–429, 1995.
J. Chen, Improvement of asymptotic formulas for the number of lattice points in a region of three dimensions. II, Sci. Sin., 12:751–764, 1963.
H. Davenport, Cubic forms in sixteen variables, Proc. R. Soc. Lond., Ser. A, 272:285–303, 1963.
H. Davenport, Analytic Methods for Diophantine Equations and Diophantine Inequalities, 2nd ed., Cambridge Univ. Press, Cambridge, 2005. (With a foreword by R.C. Vaughan, D.R. Heath-Brown, and D.E. Freeman, edited and prepared by T.D. Browning.)
N. Gafurov, On the sum of the number of divisors of a quadratic form, Dokl. Akad. Nauk Tadzh. SSR, 28:371–375, 1985.
N. Gafurov, On the number of divisors of a quadratic form, Proc. Steklov Inst. Math., 200:137–148, 1993.
R. Guo and W.G. Zhai, Some problems about the ternary quadratic form \( {m}_1^2+{m}_2^2+{m}_3^2 \), Acta Arith., 156(2):101–121, 2012.
D.R. Heath-Brown, Lattice points in the sphere, in K. Györy, H. Iwaniec, and J. Urbanowicz (Eds.), Number Theory in Progress, Vol. 2, De Gruyter, Berlin, 1999, pp. 883–892.
D.R. Heath-Brown, Cubic forms in 14 variables, Invent. Math., 170:199–230, 2007.
L.K. Hua, Some results in additive prime number theory, Q. J. Math., 9:60–80, 1938.
H.F. Liu and L.Q. Hu, On the number of divisors of a quaternary quadratic form, Int. J. Number Theory, 5:1219–1235, 2016.
C.D. Pan and C.B. Pan, Foundations of the Analytic Number Theory, Science Press, Beijing, 1991 (in Chinese).
I.M. Vinogradov, On the number of integer points in a sphere, Izv. Akad. Nauk SSSR, Ser. Mat., 27:957–968, 1963 (in Russian).
G. Yu, On the number of divisors of the quadratic form, Can. Math. Bull., 43:239–256, 2000.
M. Zhang, An asymptotic formula related to the sums of divisors, Acta Arith., 175(2):183–200, 2016.
L.L. Zhao, The sum of divisors of a quadratic form, Acta Arith., 163(2):161–177, 2014.
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*This work is supported by Natural Science Foundation of Shandong Province (grant No. ZR2018BA006) and National Natural Science Foundation of China (grant Nos. 11801328, 11771256).
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Liu, H. A Divisor Problem Attached to Regular Quadratic Forms*. Lith Math J 59, 169–184 (2019). https://doi.org/10.1007/s10986-019-09436-x
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DOI: https://doi.org/10.1007/s10986-019-09436-x