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UBC Theses and Dissertations

Prime symmetric divisor functions Woodford, Roger


In this thesis, we study a new class of divisor-related functions: the prime symmetric functions. The elementary prime symmetric functions are denned on the nonnegative integers. They take the values of the elementary symmetric functions applied to the multi-set of prime factors with repetition of an integer n. These functions are first defined i n [20]. In this thesis, we refine some of the definitions presented there, and revisit some of the key results regarding such functions. The prime symmetric functions are polynomials over Q in the elementary prime symmetric functions, with values in Z. We look at basic properties of prime symmetric functions. We study the effect of iteration of these functions, and look at the question: when does iterating produce cycles? We consider the inverse question of when and in how many ways a number n can be expressed as f(m) for certain predetermined prime symmetric functions. As well, we look at asymptotic approximations for certain classes of prime symmetric functions.

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