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arXiv · 0808.1291

The Riesz energy of the $N$-th roots of unity: an asymptotic expansion for large $N$

J. S. Brauchart·D. P. Hardin·E. B. Saff
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Abstract

We derive the complete asymptotic expansion in terms of powers of $N$ for the Riesz $s$-energy of $N$ equally spaced points on the unit circle as $N\to \infty$. For $s\ge -2$, such points form optimal energy $N$-point configurations with respect to the Riesz potential $1/r^{s}$, $s\neq0$, where $r$ is the Euclidean distance between points. By analytic continuation we deduce the expansion for all complex values of $s$. The Riemann zeta function plays an essential role in this asymptotic expansion.

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J. S. Brauchart, D. P. Hardin, E. B. Saff. 2008-08-12. The Riesz energy of the $N$-th roots of unity: an asymptotic expansion for large $N$. https://doi.org/10.1112/blms%2Fbdp034

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