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Cyclotomic polynomials at roots of unity. (English) Zbl 1435.11060
Let \(\Phi_n\) be the \(n\)th cyclotomic polynomial and let \(\xi_m\) be an arbitrary primitive \(m\)th root of unity (\(e^{2\pi ia/m)}, \,(a,m)=1\)). The paper under review evaluates \(\Phi_n(\xi_m)\) for \(1\le m\le6\) and all \(n\). These evaluations splinter into quite a few cases with some of the cases yielding rather complicated values, and we shall not state them here explicitly. To give a reader of this review a glimpse of what happens, we just mention that even the simplest case of \(\xi_1=1\) leads to three subcases: of course \(\Phi_1(1)=0\), and for \(n\ge2\), \(\Phi_n(1)=1\), except for when \(n\) is a prime power in which case \(\Phi_{p^\nu}(1)=p\). This and a closely related evaluation of \(\Phi_n(\xi_2)=\Phi_n(-1)\) are folklore background to this problem.
An earlier work of K. Motose [Bull. Fac. Sci. Technol., Hirosaki Univ. 9, No. 1, 15–27 (2006; Zbl 1193.11115)] gave evaluations of \(\Phi_n(\xi_m)\) for \(m=3, 4\) and \(6\), except that it was blemished by some error. The paper under review not only corrects the record, but obtains these evaluations by a different and more efficient technique.
As we shall explain presently, the case \(m=5\) is considerably more interesting and important. The authors’ work on this case utilizes a new general identity of independent interest which we explain next. For this identity we assume that \(n,m>1\) are coprime. Then we have
\[\Phi_n(\xi_m)=\exp\Bigl(\frac1{\varphi(m)}\sum_{\chi}C_{\chi}(\xi_m)D_{\chi}(n)\Bigr),\]
where the summation is over Dirichlet characters \(\chi\) modulo \(m\),
\[C_{\chi}(\xi_m)=\sum_{(a,m)=1}\bar{\chi}(a)\log(1-\xi_m^a) \quad\text{and}\quad D_{\chi}(n)=\chi(n)\prod_{p\mid n}(1-\bar{\chi}(p)).\]
The basic idea here is as follows. From the well-known Möbius inversion formula \(\Phi_n(z)=\prod_{d\mid n}(1-z^d)^{\mu(n/d)}\), we get \[\Phi_n(\xi_m)=\exp\Bigl(\sum_{d\mid n}\mu(n/d)\log(1-\xi_m^d)\bigr).\] The authors observe that \(\log(1-\xi_m^d)\), as a function of \(d\) with \((d,m)=1\), has a representation \[\log(1-\xi_m^d)=\frac1{\varphi(m)}\sum_{\chi}C_{\chi}(\xi_m)\chi(d),\] with the coefficients \(C_{\chi}\) computed in the standard manner. The rest of the derivation is then routine.
Utilizing the forgoing identity, the authors succeed in evaluating \(\Phi_n(\xi_5)\). As we already mentioned, this case is special, as first realized by R. C. Vaughan [Mich. Math. J. 21, 289–295 (1975; Zbl 0304.10008)]. Let \(A_n\) denote the maximum absolute value of the coefficients of \(\Phi_n\). Since degree of \(\Phi_n\) is \(\varphi(n)\), where \(\varphi\) is the Euler totient, the inequality
\[A_n\ge\frac{\max_{|z|=1}|\Phi_n(z)|}{\varphi(n)+1}\]
is immediate. Vaughan conceived of the utility of using \(\xi_5\) for manufacturing arbitrary large integers \(n\) for which \(|\Phi_n(\xi_5)|\) were particularly large, as a function of \(n\). More specifically, he proved that there exist arbitrarily large integers \(n\) for which
\[\log\log A_n>\log2\frac{\log n}{\log\log n}.\]
It follows from an earlier result of P. T. Bateman [Bull. Am. Math. Soc. 55, 1180–1181 (1949; Zbl 0035.31102)] that the constant \(\log 2\) is the best possible in this inequality. Thus the evaluation of \(\Phi_n(\xi_5)\) given in the paper under review may be viewed as a refinement of the aforementioned work of Vaughan, and a new proof of the last inequality.

MSC:
11C08 Polynomials in number theory
11R09 Polynomials (irreducibility, etc.)
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[1] B. Bzdęga, A. Herrera-Poyatos and P. Moree, Cyclotomic polynomials at roots of unity, arXiv:1611.06783 (2017).
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