A discrete spectral framework on cyclic graphs produces exact identities for special values of Dirichlet L-functions at integers and reformulates GRH for odd primitive characters via asymptotic functional equations.
The Riesz energy of the $N$-th roots of unity: an asymptotic expansion for large $N$
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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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A discrete approach to Dirichlet L-functions, their special values and zeros
A discrete spectral framework on cyclic graphs produces exact identities for special values of Dirichlet L-functions at integers and reformulates GRH for odd primitive characters via asymptotic functional equations.