## 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.

### MSC:

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

### Citations:

Zbl 1417.11028; Zbl 1464.11035
Full Text:

### References:

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