# A generalisation of theorem of Landau on sum of two squares?

Let $$r(B)$$ be the number of integers $$1 \leq n \leq B$$ such that $$n = x^2 + y^2$$ for some $$x, y \in \mathbb{Z}.$$
Then it is a known theorem of Landau that $$r(B) \sim C \frac{B}{\sqrt{\log B}}$$ for some $$C > 0$$. I was wondering if there is a generalization of this result for slightly more general function $$ax^2 + b y^2$$ where $$a, b$$ are positive integers. Thank you!

• it will turn out the same for positive form $a x^2 + b xy + c y^2,$ although the constant $C$ will change. In brief, there is a representation for some positive $n$ only when, for every prime $q$ with Legendre symbol $(b^2 - 4ac |q) = -1, \; \;$ the exponent of prime $q$ in $n = q^e \cdot other$ is even. Actually, indefinite forms should work as well, as long as the discriminant is not square Mar 19 at 15:27
• @WillJagy do you know a reference for the mentioned result? thank you Mar 19 at 15:41
• not exactly. At the end of Topics of Number Theory by LeVeque, volume 2 section 7-5, he does two squares with attention to detail. In my Dover reprint, two volumes in one, it is pages 257-263. I also have two books on analytic number theory by Apostol, might be in there. Mar 19 at 15:48

The general result you seek is due to Paul Bernays, who studied under E. Landau in Göttingen and proved in his dissertation (1912) that $$\tag{\star}\sum_{n \le x} b_Q(n) \sim C_Q \frac{x}{\sqrt{\log x}}$$ as $$x \to \infty$$ where $$Q(x,y)=ax^2+bxy+cy^2$$ is a primitive positive-definite binary quadratic form ($$b^2−4ac<0$$, $$a>0$$, $$a,b,c \in \mathbb{Z}$$) and $$b_Q$$ is the indicator function of integers representable as $$Q(x,y)$$ ($$x,y \in \mathbb{Z}$$). The constant $$C_Q$$ depends only on the discriminant of $$Q$$.
You want to apply it with $$b=0$$ (in the above notation); if $$a,b$$ (in your notation) are not coprime you need to first reduce to the primitive case before applying Bernays’ result. (E.g. if $$Q_1$$ and $$Q_2$$ are two binary quadratic form with the same discriminant, it does not necessarily follow that $$\sum_{n \le x} b_{Q_1}(n) \sim \sum_{n \le x} b_{Q_2}(n)$$ unless you first assume $$Q_i$$ are primitive.)
Since the dissertation is in German, let me mention that there are two alternative proofs, giving rate of decay for the error term in $$(\star)$$.
R. W. K. Odoni in “On norms of integers in a full module of an algebraic number field and the distribution of values of binary integral quadratic forms” (Mathematika, Lond. 22, 108–111 (1975)) proved that $$(\star)$$ holds with a relative saving of $$(\log x)^{c_Q}$$ for some $$c_Q>0$$.
O. M. Fomenko, in “Distribution of values of Fourier coefficients for modular forms of weight 1” (J. Math. Sci., New York 89, No. 1, 1051–1071 (1998)) gave a third proof, that leads to formulas for $$C_Q$$ and $$c_Q$$. See section 2 of this paper for a useful overview of other results.