Fast Calculation of the Lomb-Scargle Periodogram Using Graphics Processing Units

I introduce a new code for fast calculation of the Lomb-Scargle periodogram, that leverages the computing power of graphics processing units (GPUs). After establishing a background to the newly emergent field of GPU computing, I discuss the code design and narrate key parts of its source. Benchmarking calculations indicate no significant differences in accuracy compared to an equivalent CPU-based code. However, the differences in performance are pronounced; running on a low-end GPU, the code can match 8 CPU cores, and on a high-end GPU it is faster by a factor approaching thirty. Applications of the code include analysis of long photometric time series obtained by ongoing satellite missions and upcoming ground-based monitoring facilities; and Monte-Carlo simulation of periodogram statistical properties.Comment: Accepted by ApJ. Accompanying program source (updated since acceptance) can be downloaded from http://www.astro.wisc.edu/~townsend/resource/download/code/culsp.tar.g

Spectral exponential sums on hyperbolic surfaces I

We study an exponential sum over Laplace eigenvalues $\lambda_{j} = 1/4+t_{j}^{2}$ with $t_{j} \leqslant T$ for Maass cusp forms on $\Gamma \backslash \mathbb{H}$ as $T$ grows, where $\Gamma \subset PSL_{2}(\mathbb{R})$ is a cofinite Fuchsian group acting on the upper half-plane $\mathbb{H}$. Specifically, for the congruence subgroups $\Gamma_{0}(q), \, \Gamma_{1}(q)$ and $\Gamma(q)$, we explicitly describe each sum in terms of a certain oscillatory component, von Mangoldt-like functions and the Selberg zeta function. We also establish a new expression of the spectral exponential sum for a general cofinite group $\Gamma \subset PSL_{2}(\mathbb{R})$, and in particular we find that the behavior of the sum is decisively determined by whether $\Gamma$ is essentially cuspidal or not. We also work with certain moonshine groups for which our plotting of the spectral exponential sum alludes to the fact that the conjectural bound $O(X^{1/2+\epsilon})$ in the Prime Geodesic Theorem may be allowable. In view of our numerical evidence, the conjecture of Petridis and Risager is generalized.Comment: 20 pages, 3 figures, comments welcom

A Few Finite Trigonometric Sums

Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known, however sums with products of trigonometric functions can get complicated and may not have a simple expressions in a number of cases. Some of these sums have interesting properties and can have amazingly simple value. However, only some of them are available in literature. We obtain a number of such sums using method of residues.Comment: 11 pages, added references, corrected typo