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Partitions into prime powers and related divisor functions Mullen Woodford, Roger

Abstract

In this thesis, we will study a class of divisor functions: the prime symmetric functions. These are polynomials over Q in the so-called elementary prime symmetric functions, whose values lie in Z. The latter are defined on the nonnegative integers and take the values of the elementary symmetric functions applied to the multi-set of prime factors (with repetition) of an integer n. Initially we look at basic properties of prime symmetric functions, and consider analogues of questions posed for the usual sum of proper divisors function, such as those concerning perfect numbers or Aliquot sequences. We consider the inverse question of when, and in how many ways a number $n$ can be expressed as f(m) for certain prime symmetric functions f. Then we look at asymptotic formulae for the average orders of certain fundamental prime symmetric functions, such as the arithmetic function whose value at n is the sum of k-th powers of the prime divisors (with repetition) of n. For these last functions in particular, we also look at statistical results by comparing their distribution of values with the distribution of the largest prime factor dividing n. In addition to average orders, we look at the modular distribution of prime symmetric functions, and show that for a fundamental class, they are uniformly distributed over any fixed modulus. Then our focus shifts to the related area of partitions into prime powers. We compute the appropriate asymptotic formulae, and demonstrate important monotonicity properties. We conclude by looking at iteration problems for some of the simpler prime symmetric functions. In doing so, we consider the empirical basis for certain conjectures, and are left with many open problems.

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Attribution-NonCommercial-NoDerivatives 4.0 International

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