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Bulletin of the London Mathematical Society Advance Access originally published online on May 4, 2009
Bulletin of the London Mathematical Society 2009 41(4):621-633; doi:10.1112/blms/bdp034
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© 2009 London Mathematical Society

The Riesz energy of the Nth roots of unity: an asymptotic expansion for large N

J. S. Brauchart

Department of Analysis and Computational Number Theory (Math A)
Graz University of Technology
Steyrergasse 30
8010 Graz
Austria
brauchart@finanz.math.tu-graz.ac.at

Current address:

Center for Constructive Approximation
Department of Mathematics
Vanderbilt University
Nashville, TN 37240
USA
Johann.Brauchart@Vanderbilt.Edu

D. P. Hardin

Center for Constructive Approximation
Department of Mathematics
Vanderbilt University
Nashville, TN 37240
USA
Doug.Hardin@Vanderbilt.Edu

E. B. Saff

Center for Constructive Approximation
Department of Mathematics
Vanderbilt University
Nashville, TN 37240
USA

Received 29 August 2008.

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 -> {infty}. For s ≥ – 2, such points form optimal energy N-point configurations with respect to the Riesz potential 1/rs, s != 0, 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.

2000 Mathematics Subject Classification 30B40, 78A30 (primary), 33B15 (secondary).

The research of the first author was supported, in part, by the Austrian Science Foundation under grant S9603-N13 and by the US National Science Foundation under grant DMS-0532154 (D. P. Hardin and E. B. Saff principal investigators). The research of the second author was supported, in part, by the US National Science Foundation under grants DMS-0505756 and DMS-0532154. The research of the third author was supported, in part, by the US National Science Foundation under grants DMS-0532154 and DMS-0603828.


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