The distribution of prime numbers in short intervals

In this thesis, we focus on the problem of primes in short intervals. We will explore the main ingredients in the works by Ingham (1937), Heath-Brown and Iwaniec (1979), and Baker and Harman (1996), such as zero-density estimates, their weighted variation, and sieve methods. In Chapter 2, we will provide a generalised version of Ingham's theorem, which connects the classical zero-density estimates with primes in short intervals; our version of the theorem explicitly shows the dependence of a short interval length on the combination of the zero-free regions and the zero-density estimates. In Chapter 3, we introduce the weighted zero-density estimates due to Heath-Brown and Iwaniec, which allow one to avoid the bottlenecks of the classical approach and are used in the paper by Baker and Harman in a slightly different variation. We generalise and simplify the results by Heath-Brown and Iwaniec, explain how to switch from their variation to the variation by Baker and Harman and provide an improvement for the weighted zero-density estimates using the current classical zero-density estimates. Chapter 4 is devoted to sieve methods and Harman's comparison principle, which connects the estimates for rough numbers in short and long intervals. In addition, we prove the asymptotic for rough numbers in the sets where the prime number theorem holds --- this result allows us to shrink the length of the interval with the asymptotic for rough numbers to $x^{7/12}$. In Chapter 5, we introduce the asymptotic for the weighted sums for prime-counting functions with parameters restricted to dyadic intervals. These results are crucial for the complete understanding of the final proof by Baker and Harman, which we explain in detail in Chapter 6. Finally, we introduce an optimisation problem which, once resolved, will refine the result from the paper by Baker and Harman with the use of the generalised techniques introduced in the thesis

Some explicit results on the sum of a prime and an almost prime

Inspired by a classical result of R\'enyi, we prove that every even integer $N\geq 4$ can be written as the sum of a prime and a number with at most 369 prime factors. We also show, under assumption of the generalised Riemann hypothesis, that this result can be improved to 89 prime factors.Comment: 19 page