A Study of L-Functions: At The Edge of the Critical Strip and Within

In analytic number theory, and increasingly in other surprising places, L-functions arise naturally when describing algebraic and geometric phenomena. For example, when attempting to prove the Prime Number Theorem the values of L-functions on the one-line played a crucial role. In this thesis we discuss the theory of L-functions in two different settings.

In the classical context we provide results which give estimates for the size of a general L-function on the right edge of the critical strip, that is complex numbers with real part one. We also provide a bound for the number of zeros for the classical Riemann zeta function inside the critical strip commonly referred to as a zero density estimate.

In the second setting we study L-functions over the polynomial ring A, which is all polynomials with coefficients in a finite field of size q. As A and the ring of integers have similar structure, A is a natural candidate for analyzing classical number theoretic questions. Additionally, the truth of the Riemann Hypothesis (RH) in A yields deeper unconditional results currently unattainable over the integers. We will focus on the distribution of values of specific L-functions in two different places: On the right edge of the critical strip, that is complex numbers with real part one, and inside of the critical strip, meaning the complex numbers will have real part between one half and one.