Abstract
We study the family of self-inversive polynomials of degree \(n\), whose \(j\)th coefficient is \(\gcd (n,j)^k\), for each fixed integer \(k \ge 1\). We prove that these polynomials have all of their roots on the unit circle, with uniform angular distribution. In the process we prove some apparently new results on Jordan’s totient function. We also prove that these polynomials are irreducible, apart from an obvious linear factor, whenever \(n\) is a power of a prime, and conjecture that this holds for all \(n\).
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Acknowledgments
We thank our anonymous referee for suggesting the current substantially shortened proof of Proposition 1, improving the readability of the paper. We also thank Rob Noble of Dalhousie University for helpful discussions.
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Dedicated to the memory of Basil Gordon.
The first author was supported in part by the Natural Sciences and Engineering Research Council of Canada. The second author is grateful for the support of the Singapore MOE grant MOE2011-T2-1-090.
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Dilcher, K., Robins, S. Zeros and irreducibility of polynomials with gcd powers as coefficients. Ramanujan J 36, 227–236 (2015). https://doi.org/10.1007/s11139-014-9579-2
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DOI: https://doi.org/10.1007/s11139-014-9579-2