For \(-D<0\) a fundamental discriminant and \(f\) a binary quadratic form with discriminant \(-D\), we may ask for the number \[ N_f(x)=| \{n\leq x:n=f(m_1,m_2)\text{ for some \(m_1,m_2\in\mathbb{Z}\)}\}| \] of integers \(n\leq x\) represented by \(f\).
P. Bernays [On the representation of positive integers by primitive binary quadratic forms with a non square discriminant. (in German) Diss. Göttingen (1912; JFM 43.0278.02)] generalized Landau’s result in the case \(-D=-4\) to show that \(N_f(x)\sim \kappa_Dx/\sqrt{\log x}\) as \(x\to \infty\), where \(\kappa_D=D^{o(1)}\). However, the story for small \(x\) is quite different, and Bernay’s results do not hold until \(x>\exp(D^{1-\varepsilon})\).
Let \(h\) and \(g\) respectively denote the class number and number of genera of \(\mathbb{Q}(\sqrt{-D})\). Let \(\ell_{-D}:=L(1,\chi_{-D})(\phi(D)/D)\) and then let \[ \kappa:=\frac{\log(h/g)}{(\log 2)(\log(\ell_{-D}\log x))}. \]
This \(\kappa\) parameter is invented by the authors to measure the “transitional ranges”
\[ \exp(D^\varepsilon)\leq x\leq \exp(D^N) \]
(where \(\varepsilon>0\) is small and \(N>0\) is large).
To give more effective estimates in all ranges of \(x\), the authors suggest three ranges for estimates on \(N_f(x)\):
\[ N_f(x)\approx\begin{cases} \frac{L(1,\chi_D)}{\tau(D)}\frac{x}{\sqrt{\ell_{-D}\log x}}, &0\leq \kappa \leq 1/2,\\ \frac{L(1,\chi_{-D})}{\tau(D)}\frac{(\ell_{-D}\log x)^{-1+\kappa(1-\log(2\kappa))}}{(1+(\kappa-1/2)(1-\kappa)\sqrt{\log\log x}}, &1/2< \kappa<1,\\ \frac{x}{\sqrt{D}}, & 1\leq \kappa\ll \frac{\log D}{\log\log D}. \end{cases}\tag{1} \]
They prove all these estimates, except in the case when \(\kappa\in [1/2-\varepsilon, 1/(\log2)+\varepsilon]\) (in which case the lower bound remains to be shown). They also give and justify a heuristic argument for the three-part division.
The authors are also able to extend the results of the first author [V. Blomer, J. Reine Angew. Math. 569, 213–234 (2004; Zbl 1051.11050), J. Lond. Math. Soc. (2) 71, No. 1, 69–84 (2005; Zbl 1166.11312)] on integers \(n\leq x\) which can be written as a sum of two powerful numbers (those numbers for which \(p\mid n\) implies that \(p^2\mid n\)). In particular, they conjecture that there are
\[ \approx\frac{x(\log\log x)^{2^{2/3}-1}}{(\log x)^{1-2^{-1/3}}} \]
such integers. They are able to prove the required upper bound and can show the lower bound within a power of \(\log\log x\).
Turning to the question of counting representations, the authors give sharp bounds and asymptotics on \(\sum_{n\leq x}r_f(n)^\beta\) for \(\beta>0\), where \(r_f(n)\) is the number of inequivalent representations of \(n\) by \(f\). They obtain especially interesting results in the case that \(\beta\in\mathbb{N}\):
Corollary 1: Let \(\beta\in\mathbb{N}\) and set \(K=2^{\beta-1}\). For a given binary quadratic form \(f(y,z)=ay^2+byz+cz^2\), let \(u\) be the smallest positive integer which can be represented by some form in \(f\mathcal{G}\) (where \(\mathcal{G}\) is the subgroup of ambiguous forms). We have
\[ \sum_{n\leq x}r_f(n)^\beta=\left(a_K(\log x)^{K-1}+\frac{\pi}{\sqrt{D}}\left(1+\frac{2^{\beta-1}-1}{u}\right)\right)x \left(1+O_{\beta,\rho}((\log x)^{-\rho})\right) \]
uniformly in \(x\geq D(\log D)^{2\rho}/a\) for any \(0<\rho<1/3\) if \(\beta=2\), for any \(0<\rho<1/2\) if \(\beta=3\), and for \(0<\rho<1\) for all other \(\beta\).
Note that the authors give the \(a_K\) explicitly, but we have omitted these details in our statement of the Corollary. They furthermore obtain weak estimates for all real, positive \(\beta\). These asymptotics hold even in the intermediate range which presented a difficulty in (1).
While the proofs of the key theorems are technically elementary, they are extraordinarily intricate. As the authors remark, the results should be “useful for counting problems for binary quadratic forms.” Indeed, the asymptotics given are sharp enough to be useful. For example, results of this flavor have recently been applied to an examination of lattices with small distances by P. Moree and R. Osburn [Enseign. Math. (2) 52, 361–380 (2006; Zbl 1117.11037)].