Arithmetic Progressions of Squares and Multiple Dirichlet Series

We study a Dirichlet series in two variables which counts primitive three-term arithmetic progressions of squares. We show that this multiple Dirichlet series has meromorphic continuation to $\mathbb{C}^2$ and use Tauberian methods to obtain counts for arithmetic progressions of squares and rational points on $x^2+y^2=2$.Comment: 31 pages, (now incorporating helpful comments

The Sign of Fourier Coefficients of Half-Integral Weight Cusp Forms

From a result of Waldspurger, it is known that the normalized Fourier coefficients $a(m)$ of a half-integral weight holomorphic cusp eigenform \f are, up to a finite set of factors, one of $\pm \sqrt{L(1/2, f, \chi_m)}$ when $m$ is square-free and $f$ is the integral weight cusp form related to \f by the Shimura correspondence. In this paper we address a question posed by Kohnen: which square root is $a(m)$? In particular, if we look at the set of $a(m)$ with $m$ square-free, do these Fourier coefficients change sign infinitely often? By partially analytically continuing a related Dirichlet series, we are able to show that this is so

Detectability of gravitational wave events by spherical resonant-mass antennas

We have calculated signal-to-noise ratios for eight spherical resonant-mass antennas interacting with gravitational radiation from inspiralling and coalescing binary neutron stars and from the dynamical and secular bar-mode instability of a rapidly rotating star. We find that by using technology that could be available in the next several years, spherical antennas can detect neutron star inspiral and coalescence at a distance of 15 Mpc and the dynamical bar-mode instability at a distance of 2 Mpc.Comment: 39 pages, 4 EPS Figures, some additional SNRs for secular instabilities, some changes to LIGO SNRs, Appendix added on the asymptotic expansion of energy sensitivity, corrected supernova rates. Results available at http://www.physics.umd.edu/rgroups/gen_rel_exp/snr.html Submitted to Phys. Rev.