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## Topics in the arithmetic of polynomials over finite fields

Afshar, Ardavan; (2020) Topics in the arithmetic of polynomials over finite fields. Doctoral thesis (Ph.D), UCL (University College London).

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## Abstract

In this thesis, we investigate various topics regarding the arithmetic of polynomials over finite fields. In particular, we explore the analogy between the integers and this polynomial ring, and exploit the additional structure of the latter in order to derive arithmetic statistics which go beyond what can currently be proved in the integer setting. First, we adapt the Selberg-Delange method to prove an asymptotic formula for counting polynomials with a given number of prime factors. We then extend this formula to cases in which these polynomials are restricted first to arithmetic progressions, and then to `short intervals'. In both cases, we obtain better ranges for the associated parameters than in the integer setting, by using Weil's Riemann Hypothesis for curves over finite fields. Then, we investigate highly composite polynomials and the divisor function for polynomials over a finite field, as inspired by Ramanujan's work on highly composite numbers. We determine a family of highly composite polynomials which is not too sparse, and use it to compute the maximum order of the divisor function up to an error which is much smaller than in the case of integers, even when the Riemann Hypothesis is assumed there. Afterwards, we take a brief aside to discuss the connection between the Generalised Divisor Problem and the Lindelöf Hypothesis in the integer setting. Next, we prove that for a certain set of multiplicative functions on the polynomial ring, the bound in Halász's Theorem can be improved. Conversely, we determine a criterion for when the general bound is actually attained, and construct an example which satisfies this criterion. Finally, in the other direction, we develop a formula for the Möbius function of a number field which is related to Pellet's Formula for the Möbius function of the polynomial ring.