On asymptotic distributions of arithmetical functions

Abstract

1. The present note was suggested by recent work of H. Davenport, [3 ],t S. Bochner and B. Jessen, [2 ], and A. Wintner and B. Jessen, [6 ]. Davenport established the existence of asymptotic distribution functions for a certain class of arithmetical functions by an extension of a method previously used by the author, [8], [9], in a similar investigation. This method was based on the consideration of the moments of the distribution functions. In questions of asymptotic distribution, however, Bochner and Jessen have shown the great advantage of dealing directly with the Fourier transforms of the distribution functions. This advantage becomes again apparent if the method of Fourier transforms, whose adaptation to sequences is fully developed in ?I, is applied to Davenport's problem. This is precisely what we shall do in ?II; the result thus obtained (Theorem 1) insures the existence of the asymptotic distribution function for a; very large class of (positive and multiplicative) arithmetical functions. It includes Davenport's and the author's previous results and yields readily (by suitable specializations of the arithmetical function involved) the frequencies of certain classes of integers investigated by W. Feller and E. Tornier, [4], in an entirely different way. The connectiQn with the work of Wintner and Jessen, [6], is as follows. The distribution function co(x) =X(eu) of Theorem 1 is a special example of the infinite convolutions of purely discontinuous distribution functions investigated by these authors. They have shown (1[6], Theorem 35) that such infinite convolutions can be only either purely discontinuous or else everywhere continuous, and in the latter case either singular functions or else absolutely continuous functions. These general results apply immediately to our special situation, but new and probably difficult problems arise which may be mentioned here. Theorem 1 gives simple sufficient conditions to insure continuity or discontinuity of co(x); the problem of finding similar necessary and sufficient conditions for continuity remains unsolved. The more delicate problem of deciding whether a continuous c(x) is singular or absolutely