Estimates for representation numbers of quadratic forms
type: Item
instance of: scholarly article
title: Estimates for representation numbers of quadratic forms
zbMATH Open document ID: 1135.11020
DOI: 10.1215/S0012-7094-06-13522-6
author: Valentin Blomer
author: Andrew Granville
published in: Duke Mathematical Journal
publication date: 2006-12-05T00:00:00Z
review text: 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 \(f\). \textit{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 [\textit{V. Blomer}, J. Reine Angew. Math. 569, 213--234 (2004; Zbl 1051.11050), …
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 11N56
zbMATH DE Number: 5077260
zbMATH Keywords: binary quadratic form
zbMATH Keywords: representation number
zbMATH Keywords: powerful number
reviewed by: Scott Duke Kominers
MaRDI profile type: Publication
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full work available at URL: s0012-7094-06-13522-6
OpenAlex ID: W2061029499
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description: scientific article; zbMATH DE number 5077260