This research concerns additive and multiplicative functions in number theory. They will be studied by augmenting the classical methods of Fourier analysis, sieve methods and exponential sums. New ideas will be taken from functional analysis to apply to their study. The functional analysis is to be carried out in an infinite collection of finite dimensional spaces, in a uniform manner. Besides allowing a re- interpretation of certain well-known results, such as the Turan- Kubilius inequality, this approach offers methods and directions for future development. Most of the classical functions in number theory, such as the number of divisors of an integer or the number of natural numbers less than an integer which are relatively prime to that integer, are what are called multiplicative functions. This research will apply various tools from analysis and probability theory to study the general properties of such functions and the closely related functions called additive functions.
