Explicit Estimates in the Theory of Prime Numbers

TL;DR

This thesis provides explicit bounds and results in prime number theory, including prime distribution in short intervals, additive representations involving primes, and Ramanujan's inequality, advancing the field with concrete, verifiable estimates.

Contribution

It introduces new explicit bounds for primes in short intervals, proves additive number theory results involving primes and square-free numbers, and refines Ramanujan's inequality with explicit constants.

Findings

Existence of primes between consecutive cubes for large n

Primes in short intervals assuming Riemann Hypothesis

Every integer > 2 as sum of a prime and a square-free number

Abstract

It is the purpose of this thesis to enunciate and prove a collection of explicit results in the theory of prime numbers. First, the problem of primes in short intervals is considered. We prove that there is a prime between consecutive cubes $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.3))$. To prove this, we first derive an explicit version of the Riemann--von Mangoldt explicit formula. We then assume the Riemann hypothesis and show that there will be a prime in the interval $(x-4/ \pi \sqrt{x} \log x, x]$ for all $x > 2$. Moreover, we show that the constant $4/\pi$ can be reduced to $(1+\epsilon)$ for all sufficiently large values of $x$. Using explicit results on primes in arithmetic progressions, we prove two new results in additive number theory. First, we prove that every integer greater than 2 can be written as the sum of a prime and a square-free number. We then work similarly to prove that every integer greater than 10 and not congruent to $1$ modulo $4$ can be written as the sum of the square of a prime and a square-free number. Finally, we provide new explicit results on an arcane inequality of Ramanujan.