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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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##### References:
 [1] B. Bzdęga, A. Herrera-Poyatos and P. Moree, Cyclotomic polynomials at roots of unity, arXiv:1611.06783 (2017). [2] F.-E. Diederichsen, Über die Ausreduktion ganzzahliger Gruppendarstellungen bei arithmetischer Äquivalenz, Abh. Math. Sem. Hansischen Univ. 13 (1940), 357-412. [6] G. Dresden, Resultants of cyclotomic polynomials, Rocky Mountain J. Math. 42 (2012), 1461-1469. · JFM 66.0089.01 [3] H. M. Edwards, Fermat’s Last Theorem. A Genetic Introduction to Algebraic Number Theory, Grad. Texts in Math. 50, Springer, New York, 1977. [8] E. Lehmer, A numerical function applied to cyclotomy, Bull. Amer. Math. Soc. 36 (1930), 291-298. [4] S. Louboutin, Resultants of cyclotomic polynomials, Publ. Math. Debrecen 50 (1997), 75-77. · Zbl 0881.11073 [5] K. Motose, On values of cyclotomic polynomials. VIII, Bull. Fac. Sci. Technol. Hirosaki Univ. 9 (2006), 15-27. · Zbl 1193.11115 [6] R. Thangadurai, On the coefficients of cyclotomic polynomials, in: Cyclotomic Fields and Related Topics (Pune, 1999), Bhaskaracharya Pratishthana, Pune, 2000, 311-322. [12] R. C. Vaughan, Bounds for the coefficients of cyclotomic polynomials, Michigan Math. J. 21 (1975), 289-295.
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