The periodic zeta covariance function for Gaussian process regression
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I consider the Lerch-Hurwitz or periodic zeta function as covariance function of a periodic continuous-time stationary stochastic process. The function can be parametrized with a continuous index ν which regulates the continuity and differentiability properties of the process in a way completely analogous to the parameter ν of the Matérn class of covariance functions. This makes the periodic zeta a good companion to add a power-law prior spectrum seasonal component to a Matérn prior for Gaussian process regression. It is also a close relative of the circular Matérn covariance, and likewise can be used on spheres up to dimension three. Since this special function is not generally available in standard libraries, I explain in detail the numerical implementation.
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