On the representation of integers by cyclotomic forms with large degree. (English) Zbl 1455.11066

This paper develops the theme begun in the authors’ previous work [Acta Arith. 184, No. 1, 67–86 (2018; Zbl 1417.11028)]. Let \(d\ge 4\) be a value assumed by the Euler \(\phi\)-function, and let \(A_d(N)\) be the number of positive integers \(m\le N\) that are representable as a value \(m=\Phi_n(x,y)\) of a cyclotomic form of degree \(\phi(n)\ge d\), using integers \(x,y\) with \(\max(|x|,|y|)\ge 2\).
Applying methods from the earlier paper it is shown that \[A_k(N)\ll N^{2/k}(\log N)^{1, 611},\] with an implied constant independent of \(k\). This estimate may be applied to degrees greater than \(2d\), say. Thus to give an asymptotic formula for \(A_d(N)\) it suffices to consider the finitely many forms \(\Phi_n\) with \(d\le \phi(n)\le 2d\). This question is handled using a result of C. L. Stewart and S. Y. Xiao [Math. Ann. 375, No. 1–2, 133–163 (2019; Zbl 1464.11035)], the outcome being that \[A_d(N)= C_d N^{2/d}+O_d(N^{2/d^{\dagger}})+O_{d,\varepsilon}(N^{\eta_d+\varepsilon}),\] where \(d^{\dagger}\ge d+2\) is defined to be the next value assumed by \(\phi\) after \(d\); and \(\eta_d\), coming ultimately from the determinant method, is given by \[\eta_d=\left\{\begin{array}{ll}\{1/2+9/4\sqrt{d}\}/d, & (4\le d\le 20),\\ 1/d, & d\ge 21,\end{array}\right.\] so that \(\eta_d<2/d\) in all cases.


11E76 Forms of degree higher than two
11D45 Counting solutions of Diophantine equations
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