The wave nature of continuous gravitational waves from microlensing

Gravitational wave predicted by General Relativity is the transverse wave of spatial strain. Several gravitational waveform signals from binary black holes and from a binary neutron star system accompanied by electromagnetic counterparts have been recorded by advanced LIGO and advanced Virgo. In analogy to light, the spatial fringes of diffraction and interference should also exist as the important features of gravitational waves. We propose that observational detection of such fringes could be achieved through gravitational lensing of continuous gravitational waves. The lenses would play the role of the diffraction barriers. Considering peculiar motions of the observer, the lens and the source, the spatial amplitude variation of diffraction or interference fringes should be detectable as an amplitude modulation of monochromatic gravitational signal.Comment: Accepted for publication in The Astrophysical Journa

Rough Numbers and Variations on the Erdős--Kac Theorem

The study of arithmetic functions, functions with domain N and codomain C, has been a central topic in number theory. This work is dedicated to the study of the distribution of arithmetic functions of great interest in analytic and probabilistic number theory. In the first part, we study the distribution of positive integers free of prime factors less than or equal to any given real number y\u3e=1. Denoting by Phi(x,y) the count of these numbers up to any given x\u3e=y, we show, by a combination of analytic methods and sieves, that Phi(x,y)\u3c0.6x/\log y holds uniformly for all 3\u3c=y\u3c=sqrt{x}, improving upon an earlier result of the author in the same range. We also prove numerically explicit estimates of the de Bruijn type for Phi(x,y) which are applicable in wide ranges. In the second part, we turn to the topic of weighted Erdős--Kac theorems for general additive functions. Our results concern the distribution of additive functions f(n) weighted by nonnegative multiplicative functions alpha(n) in a wide class. Building on the moment method of Granville, Soundararajan, Khan, Milinovich and Subedi, we establish uniform asymptotic formulas for the moments of f(n) with a suitable growth rate. Our method also enables us to prove a qualitative result on the moments which extends a theorem of Delange and Halberstam on the moments of additive functions. As a consequence, we obtain a weighted analogue of the Kubilius--Shapiro theorem with simple and interesting applications to the Ramanujan tau function and Euler\u27s totient function, the latter of which generalizes an old result of Erdős and Pomerance which shows that as an arithmetic function, the total number of prime factors of values of Euler\u27s totient function satisfies a Gaussian law