## Duke Mathematical Journal

### Estimates for representation numbers of quadratic forms

#### Abstract

Let $f$ be a primitive positive integral binary quadratic form of discriminant $-D$, and let $r_f(n)$ be the number of representations of $n$ by $f$ up to automorphisms of $f$. In this article, we give estimates and asymptotics for the quantity $\sum_{n \leq x} r_f(n)^{\beta}$ for all $\beta \geq 0$ and uniformly in $D = o(x)$. As a consequence, we get more-precise estimates for the number of integers which can be written as the sum of two powerful numbers

#### Article information

Source
Duke Math. J., Volume 135, Number 2 (2006), 261-302.

Dates
First available in Project Euclid: 17 October 2006

https://projecteuclid.org/euclid.dmj/1161093265

Digital Object Identifier
doi:10.1215/S0012-7094-06-13522-6

Mathematical Reviews number (MathSciNet)
MR2267284

Zentralblatt MATH identifier
1135.11020

Subjects
Primary: 11E16: General binary quadratic forms
Secondary: 11N56: Rate of growth of arithmetic functions

#### Citation

Blomer, Valentin; Granville, Andrew. Estimates for representation numbers of quadratic forms. Duke Math. J. 135 (2006), no. 2, 261--302. doi:10.1215/S0012-7094-06-13522-6. https://projecteuclid.org/euclid.dmj/1161093265

#### References

• N. C. Ankeny and S. Chowla, The relation between the class number and the distribution of primes, Proc. Amer. Math. Soc. 1 (1950), 775--776.
• P. Bernays, Über die Darstellung von positiven, ganzen Zahlen durch die primitiven, binären quadratischen Formen einer nicht quadratischen Diskriminante, Ph.D. dissertation, Georg-August-Universität, Göttingen, Germany, 1912.
• V. Blomer, Binary quadratic forms with large discriminants and sums of two squareful numbers, J. Reine Angew. Math. 569 (2004), 213--234.; $\hbox\it II$, J. London Math. Soc. (2) 71 (2005), 69--84. $\!$;
• —, On cusp forms associated with binary theta series, Arch. Math. (Basel) 82 (2004), 140--146.
• J. BrüDern, Einführung in die analytische Zahlentheorie, Springer, Berlin, 1995.
• D. A. Cox, Primes of the Form $x^2+ny^2$: Fermat, Class Field Theory and Complex Multiplication, Wiley-Intersci. Publ., Wiley, New York, 1989.
• P. ErdöS and A. SáRköZy, On the number of prime factors of integers, Acta Sci. Math. (Szeged) 42 (1980), 237--246.
• E. Fogels, On the zeros of Hecke's L-functions, I, Acta Arith. 7 (1961/1962), 87--106.
• A. Hildebrand, On the number of positive integers $\leq x$ and free of prime factors $>y$, J. Number Theory 22 (1986), 289--307.
• H. Iwaniec, The generalized Hardy-Littlewood's problem involving a quadratic polynomial with coprime discriminants, Acta Arith. 27 (1975), 421--446.
• R. Murty And R. Osburn, Representations of integers by certain positive definite binary quadratic forms, preprint.
• G. Pall, The structure of the number of representations function in a positive binary quadratic form, Math. Z. 36 (1933), 321--343.