Explicit Estimates in the Theory of Prime Numbers

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 furnish an explicit result on the existence of primes between consecutive cubes. To prove this, we first derive an explicit version of the Riemann--von Mangoldt explicit formula. We then assume the Riemann hypothesis and improve on the known conditional explicit estimates for primes in short intervals. Using recent results on primes in arithmetic progressions, we prove two new results in additive number theory. First, we prove that every integer greater than two 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 ten and not congruent to one modulo four 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. We solve the inequality completely on the assumption of the Riemann hypothesis