Cyclotomic polynomial coefficients \(a(n,k)\) with \(n\) and \(k\) in prescribed residue classes
Let \(n\geq 1\) be an integer. Let \(\Phi_n(x)=\sum_{k=0}^{\varphi(n)} a(n,k) x^k\) denote the \(n\)th cyclotomic polynomial. Set \(a(n,k)=0\) for \(k>\varphi(n)\). The coefficients \(a(n,k)\) have a strong tendency to be small, e.g., for \(n<105\) one has \(a(n,k)\in \{-1,0,1\}\). Nevertheless it can be shown that \textit{every} integer can occur as a cyclotomic coefficient. In this paper the author determines the set \(\{a(n,k)\}\) as \(n\) runs through the positive integers congruent to \(a\pmod d\) and \(k\) runs through the non-negative integers congruent to \(b\pmod f\), with \(0\leq a<d\) and \(0\leq b<f\). Denote this set by \(S(a,d;b,f)\). Define the squarefree integer \(t(n)\) to be the product of the distinct primes dividing \(n\), and set \(s(n)=n/t(n)\). The author's main result is that if \(\gcd(s(\text{gcd}(a,d)),f)\) divides \(b\), then \(S(a,d;b,f)=\mathbb Z\) and \(S(a,d;b,f)=\{0\}\) otherwise. The proof uses the quantitative form of Dirichlet's prime number theorem for arithmetic progressions. \textit{C. G. Ji, W.-P. Li} and the reviewer [Discrete Math. 309, N. 6, 1720--1723 (2009; Zbl 1221.11067)] proved that \(S(0,d;b,1)=\mathbb Z\), a special case of the author's result. Consider the power series expansion of \(\Phi_n(x)^{-1}\) at \(x=0\): \(\Phi_n(x)^{-1}=\sum_{k=0}^{\infty}c(n,k)x^k\). It is easy to see that the \(c(n,k)\) are integers. The author also shows that her main result remains true if one replaces \(a(n,k)\) by \(c(n,k)\). For a detailed study of the coefficients \(c(n,k)\) see the reviewer [J. Number Theory 129, No. 3, 667--680 (2009; Zbl 1220.11037)].
scientific article; zbMATH DE number 5931159
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