Elliot 9530690 This award provides funding for an investigation that will incorporate a drive torwards a 1904 conjecture of Dickson, and the celebrated prime-pair problem. This drive will be developed using harmonic analysis on the multiplicative group of the rationals. To this end, unimodular multiplicative functions are to be viewed as characters on the group, and a systematic study made of their value distribution on arithmetic progressions with large differences. Besides their intrinsic interest and applicability in number theory, the methods to be employed offer a theoretical approach to various algorithms used in the fast factorization of large numbers; hence, these methods bear on public encryption. Methods already developed for the study of arithmetic functions, exposed in part in an AMS Memoir, more fully in a book to be published by Cambridge University Press, are to be employed to radically restructure the teaching of Analytic Number Theory in the proposer's university, and, in particular, applied to the training of several graduate students working towards a Ph. D. An auxiliary study is to be made of the value distribution of the least primitive root of a prime modulus. The function is connnected to an old conjecture of Vinogradov, and of vital interest in the theory of computing. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.
