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BINARY QUADRATIC FORMS WITH LARGE DISCRIMINANTS AND SUMS OF TWO SQUAREFUL NUMBERS II

Published online by Cambridge University Press:  04 February 2005

VALENTIN BLOMER
VALENTIN BLOMER
Affiliation:
University of Toronto, Department of Mathematics, 100 St George Street, Toronto, Ontario, Canada M5S 3G3 vblomer@math.toronto.edu
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Abstract

Let $\textbf{F}\,{=}\,(F_1, \ldots, F_m)$ be an $m$-tuple of primitive positive binary quadratic forms and let $U_{\textbf{F}}(x)$ be the number of integers not exceeding $x$ that can be represented simultaneously by all the forms $F_j$, $j = 1, \ldots, m$. Sharp upper and lower bounds for $U_{\textbf{F}}(x)$ are given uniformly in the discriminants of the quadratic forms.

As an application, a problem of Erdős is considered. Let $V(x)$ be the number of integers not exceeding $x$ that are representable as a sum of two squareful numbers. Then $V(x) = x(\log x)^{-\alpha+o(1)}$ with $\alpha\,{=}\,1-2^{-1/3}\,{=}\,0.206\ldots$.

Keywords

11N2511E1611E2511N37
Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 71 , Issue 1 , February 2005 , pp. 69 - 84
Copyright
The London Mathematical Society 2005

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