Computing Stieltjes constants using complex integration
The Stieltjes constants $\gamma\_n$ are the coefficients appearing in the Laurent series of the Riemann zeta function at $s = 1$. We give a simple and efficient method to compute a $p$-bit approximation of $\gamma\_n$ with rigorous error bounds. Starting from an integral representation due to Blagouchine, we shift the contour to eliminate cancellation. The integral is then evaluated numerically in ball arithmetic using the Petras algorithm, with the use of a Taylor expansion for bounds near the saddle point. This appears to be the first algorithm for Stieltjes constants with uniformly low complexity with respect to both $n$ and $p$. An implementation is provided in the Arb library. We can, for example, compute $\gamma\_n$ to 1000 digits in a minute for any $n \le 10^{100}$.
Power series expansions of Riemann’s function
We show how high-precision values of the coefficients of power series expansions of functions related to Riemann's í function may be calculated. We also show how the Stieltjes constants can be evaluated using this scheme and how the Riemann hypothesis can be expressed in terms of the behavior of two of the sequences of coefficients. High-precision values for the coefficients of these power series are found using Mathematica TM .
Noctilucent cloud observations over Greenland by a Rayleigh lidar
We report on the detection of noctilucent clouds (NLCs) by a Rayleigh lidar system located on the west coast of Greenland during operations in July and August 1994. The lidar detected NLCs in the zenith on three nights (July 28–29, July 31–August 1, and August 4). The lidar measurements are presented at temporal (1 minute) and spatial (192 meter) resolutions exceeding previously reported lidar measurements of NLCs and reveal significantly more structure in the clouds. Unique cloud features are observed for each night NLCs were detected. Because of the large backscatter ratio associated with NLCs, the volume backscatter coefficient for the cloud particles can be determined more accurately from the lidar signal than the backscatter ratio as the upper mesospheric molecular number density is negligible in the volume backscatter coefficient calculation. The volume backscatter coefficient values determined from the three nights of NLC observations would be classified by current models as moderate to extreme. The extreme backscatter from NLC measurements on August 4 tests the assumptions used in existing models.