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Abstract
This thesis discusses three topics related to the distributions of arithmetic functions. The first topic is the distribution function of a polynomial of additive functions. Roughly speaking, the distribution function of an arithmetic function f records how often f lies below a given value. We show that certain polynomials of additive functions with continuous distribution functions also have continuous distribution functions.The second topic is the range of Euler's totient function. For an irreducible quadratic polynomial P, we prove that for almost all n, the equation phi(m) = P(n) has no solutions.The final topic is additively unique sets of primes. A set S is additively unique if the only multiplicative functions possessing a certain invariant on S are f(n) = 0 and f(n) = n. We classify the additively unique sets of primes.