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## Some topics in the analytic number theory of polynomials over a finite field

Porritt, Sam; (2020) Some topics in the analytic number theory of polynomials over a finite field. Doctoral thesis (Ph.D), UCL (University College London).

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## Abstract

There are striking similarities between the ring of integers and the ring of polynomials in one variable over a finite field. This thesis explores some of these similarities from an analytic number theoretic perspective. It develops a polynomial analogue of techniques for extracting number theoretic information from analytic functions known as the Selberg--Delange method. A motivating problem for the original development of this theory was the problem of counting integers with a prescribed number of prime factors. After presenting the theory in the context of counting polynomials with a prescribed number of prime factors in arithmetic progressions and short intervals, a refined version of the method is presented to study some related quantities in more detail. This work has applications to the study of so-called prime number races questions for polynomials with a prescribed number of prime factors. As a prelude to this work on the Selberg--Delange method, an application from the integer version is given. It concerns the distribution of the values of $\omega(n)$, the number of prime divisors of $n$, in different residue classes. We also prove some results concerning the existence and number of prime polynomials whose coefficients satisfy certain conditions. These can be compared with results about the existence and number of prime numbers whose digits satisfy certain conditions. In particular, we study prime polynomials whose coefficients are restricted to a given subset of the underlying finite field and those whose coefficients satisfy a given linear equation. These results make use of additive characters and as prelude to them, a result concerning the correlation of the polynomial analogue of the exponential function with the multiplicative M\"{o}bius function is presented.