The Lind-Lehmer constant for cyclic groups of order less than 892,371,480 (Q744833)

Let \(\mathcal{M}_n\) be the minimum over \(M_n(F)\), where \(M_n(F)=\prod_{j=1}^n F(e^{2\pi i j/n})\) and \(F\) runs through all polynomials with integer coefficients for which \(|M_n(F)|>1\). It is known that \(\mathcal{M}_n=2\) for \(n\) odd. For \(n\) even, \textit{N. Kaiblinger} showed earlier some upper and lower bounds for \(\mathcal{M}_n\) and determined the exact value for many values of \(n\) [Acta Arith. 142, No. 1, 79--84 (2010; Zbl 1197.11032)]. The first undetermined value is \(\mathcal{M}_{420}\), although it was known that \(\mathcal{M}_{420} \in \{8,9,11\}\). In this paper, the authors determine the value of \(\mathcal{M}_{420}\) by showing that \(\mathcal{M}_{n}=11\) if \(n=2^2 \cdot 3 \cdot 5 \cdot 7 \cdot m\), where \(m\) is not divisible by \(11\). In a similar fashion the authors establishe that \(\mathcal{M}_{n}=13\) for \(n=2^2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot m\), where \(m\) is not divisible by \(13\), etc. up to \(\mathcal{M}_{n}=23\) for \(n=2^3 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13\cdot 17 \cdot 19 \cdot m\), where \(m\) is not divisible by \(23\). The first undetermined value is now \(\mathcal{M}_{2^3 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13\cdot 17 \cdot 19 \cdot 23} \in \{25,27\}\).