## University Scholar Projects

#### Date of Completion

Spring 5-1-2021

Keith Conrad; Alvaró Lozano-Robledo; Benjamin Fuller

Mathematics

Computer Science

#### Disciplines

Analysis | Information Security | Number Theory | Theory and Algorithms

#### Abstract

The Riemann Hypothesis, posed in 1859 by Bernhard Riemann, is about zeros
of the Riemann zeta-function in the complex plane. The zeta-function can be repre-
sented as a sum over positive integers n of terms 1/ns when s is a complex number
with real part greater than 1. It may also be represented in this region as a prod-
uct over the primes called an Euler product. These definitions of the zeta-function
allow us to find other representations that are valid in more of the complex plane,
including a product representation over its zeros. The Riemann Hypothesis says that
all zeros of the zeta-function with real part between 0 and 1 fall exactly on the line
Re(s) = 1/2.
The Generalized Riemann Hypothesis deals with a similar class of functions to the
zeta-function called Dirichlet L-functions. This time, instead of a series with terms
1/ns, we consider the series with terms χ(n)/ns for a (primitive) Dirichlet character
χ. Similar to the zeta-function, this definition of a Dirchlet L-function leads to other
representations, including an Euler product over the primes and a Hadamard product
over its zeros. The Generalized Riemann Hypothesis says that all zeros of a Dirichlet
L-function with real part between 0 and 1 have real part 1/2. Comparing product
representations of the zeta-function and L-functions over prime numbers and over
zeros gives intuition as to why the Riemann Hypothesis and Generalized Riemann
Hypothesis about zeros of certain functions have implications for the prime numbers.
In this thesis we look at concepts necessary to build up an understanding of the
Generalized Riemann Hypothesis (including the zeta-function, Dirichlet characters,
and some background from complex analysis) and then discuss one application: pri-
mality testing. Specifically, we will show how the Generalized Riemann Hypothesis
implies a widely used probabilistic primality test could be turned into an efficient,
usable, deterministic primality test.

COinS