Date of Award

8-2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

Committee Chair/Advisor

Neil Calkin

Committee Member

William Bridges

Committee Member

Matthew Saltzman

Committee Member

Keri Ann Sather-Wagstaff

Abstract

This dissertation explores fundamental conjectures in number theory, focusing on the distribution patterns of representation functions in prime pairs. The work concentrates on twin primes, cousin primes, and primes separated by six units, offering a fresh heuristic interpretation of the Hardy-Littlewood correction factor. The analysis progresses to investigate the partition function for prime pairs in the form $(p, p+k)$, specifically for $k = 2, 4, 6$. The study culminates in the derivation of a general formula for prime pairs $(p, p+d)$, where $d$ is an even integer. Drawing on the insights gleaned from examining the correction factor, this dissertation proposes new conjectures about the partition functions of these distinct prime pairs.

Included in

Number Theory Commons

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