Improving an inequality for the divisor function

We improve using elementary means an explicit bound on the divisor function due to Friedlander and Iwaniec. Consequently we modestly improve a result regarding a sieving inequality for Gaussian sequences.Comment: 8 pages, to appear in Bull. AustM

The temporal power spectrum of atmospheric fluctuations due to water vapor

Irregular variations in the refractivity of the atmosphere cause fluctuations in the phase measured by interferometers, limiting the spatial resolution that can be obtained. For frequencies up to the far infrared, water vapor is the dominant cause of the variations. The temporal power spectrum of the phase fluctuations is needed to assess correction schemes such as phase referencing using a nearby calibrator and water vapor radiometry. A model is developed for the temporal power spectrum of phase fluctuations measured by an interferometer through a layer of Kolmogorov turbulence of arbitrary thickness. It is found that both the orientation of the baseline with respect to the wind direction and the elevation of the observations can have a large effect on the temporal power spectrum. Plots of the spectral density distribution, where the area under the curve is proportional to phase power, show that substantial contributions from length scales as long as 100 times the interferometer baseline are possible. The model is generally consistent with data from the 12-GHz phase monitor at the Owens Valley Radio Observatory, and allows the data to be extrapolated to an arbitrary baseline, observing frequency and elevation. There is some evidence that there can be more than one component of turbulence present at a given time for the Owens Valley. The validity of the frozen turbulence assumption and the geometrical optics approximation is discussed and found to be reasonable under most conditions. The models and data presented here form the basis of an analysis of phase calibration and water vapor radiometry (Lay 1997)

Phase calibration and water vapor radiometry for millimeter-wave arrays

Correcting for the fluctuations in atmospheric path length caused by water vapor is a major challenge facing millimeter- and submillimeter-wave interferometers, and one that must be overcome to obtain routine sub-arcsecond resolution. Using the model for the power spectrum of phase fluctuations developed in Lay (1997), the existing technique of phase referencing to a bright calibrator object is analysed. It is shown that the phase errors after calibration have comparable contributions from both the target and calibrator measurements. The technique of water vapor radiometry, where the amount of emission from water vapor in the beam of each antenna is used to estimate a path correction, is also examined. It is found that there are two levels on which a correction can be made. The simplest corrects just the fluctuations within each on-source period; the calibration requirements for the radiometers are modest, and this partial correction can give a substantial improvement in the resolution and coherence time of an interferometer. The atmospheric fluctuations on longer timescales remain uncorrected, however, and are significant. To remove these, a full correction is required, which measures the change in the path difference that occurs when moving between the calibrator and the target, in addition to the on-source fluctuations. Since there can be a large difference in airmass between the calibrator and the target, measuring this change requires that the radiometers have the same response to a given column of water vapor to within ~0.1 %. Two possible methods of achieving this very stringent limit are outlined. For reasonable observing conditions at 230 GHz, it is predicted that the effective atmospheric "seeing" (the apparent smearing of the sky brightness distribution due to the atmosphere) is improved from 0.6" (phase referencing every 25 minutes) to 0.3" (phase referencing and partial radiometric correction). A full radiometric correction would, in principle, restore perfect seeing