Abstract
One of the groundbreaking results of 19th century mathematics is the prime number theorem. It states the amount of primes less than a number x, denoted π(x), behaves similarly to the function x / log x as x increases. The exact nature of this relationship is that of what is referred to asymptotic equivalence. The goal is to prove this result using modern methods arising from Newman and his Tauberian Theorem. Presented is an indirect approach, that makes use of the observed dependence of the Prime Number Theorem on particular properties of simpler, more approachable functions. Following this proof, there is discussion and exploration on the relationship of the Riemann zeta function and the Prime Number Theorem. As we find, the value of this zeta function is dependent on π(x). Further, the prime counting function can be given an explicit, algebraic definition using the Riemann zeta and its cousins. Throughout both of these discussions, extra care is taken to address the underlying principles of analysis that are often taken for granted.
Advisor
Bush, Michael
Department
Mathematics
Recommended Citation
Jordan, Thomas J., "Proving Prime Number Theorem and Its Relationship to the Riemann Zeta Function" (2024). Senior Independent Study Theses. Paper 10944.
https://openworks.wooster.edu/independentstudy/10944
Disciplines
Analysis | Number Theory
Keywords
Prime numbers, Prime Number Theorem, zeta, counting functions, Tauberian Theorems
Publication Date
2024
Degree Granted
Bachelor of Arts
Document Type
Senior Independent Study Thesis
© Copyright 2024 Thomas J. Jordan