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Linear relations amongst sums of two squares. (English) Zbl 1161.11387
Reid, Miles (ed.) et al., Number theory and algebraic geometry. To Peter Swinnerton-Dyer on his 75th birthday. Cambridge: Cambridge University Press (ISBN 0-521-54518-8/pbk). London Mathematical Society Lecture Note Series 303, 133-176 (2003).
Consider the sums of two squares. It is easy to show that this set contains infinitely many arithmetic progressions of 4 distinct integers. The question of frequency of such progressions is addressed.
Let \(r(n)\) denote the number of representations of \(n\) as a sum of two squares. Then the author shows that \[ \sum_{a<b<c<d\leq X} r(a)r(b)r(c)r(d)=CX^2+E(X), \] where \(a\), \(b\), \(c\), \(d\) are restricted to arithmetic progressions of length 4, \(E(X)\ll X^ 2\log^\delta\,X\) for some explicitly given \(\delta>1/25\), and \(C>0\) is a constant.
The result is also related to the problem of counting rational points on varieties of the type \( L_ 1(x_ 1,x_ 2)L_ 2(x_ 1,x_ 2)=x_ 3^ 2+x_ 4^ 2\), \(L_ 3(x_ 1,x_ 2)L_ 4(x_ 1,x_ 2)=x_ 5^ 2+x_ 6^ 2\).
Moreover estimates of the sum \[ \sum_{x\in{\mathcal R}_2}r\bigl(L_ 1({\mathbf x})L_2({\mathbf x})\bigr)r\bigl(L_ 3({\mathbf x})L_ 4({\mathbf x})\bigr) \] are proved. Several interesting choices of the forms \(L_ i\) are investigated in more detail.
For the entire collection see [Zbl 1050.11004].

11N25 Distribution of integers with specified multiplicative constraints
11E25 Sums of squares and representations by other particular quadratic forms
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