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!
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3$\begingroup$ 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 $\endgroup$– Will JagyMar 19 at 15:27
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$\begingroup$ @WillJagy do you know a reference for the mentioned result? thank you $\endgroup$– Johnny T.Mar 19 at 15:41
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2$\begingroup$ 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. $\endgroup$– Will JagyMar 19 at 15:48
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1 Answer
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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.)
The dissertation is titled “Über die Darstellung von positiven, ganzen Zahlen durch die primitiven, binären quadratischen Formen einer nicht-quadratischen Diskriminante”, and is available here.
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.
answered Mar 19 at 22:07