On zero-density estimates and primes in short intervals

Wednesday, 28 June 2023

Abstract

I would like to introduce the main methods used to prove there exists a prime number in the interval $[x – x^{\theta}, x]$ for $0 < \theta < 1$ (which we will refer to as a short interval). The main idea of all these methods is to connect the prime-counting function with some Dirichlet polynomials, for which we can derive the asymptotic or lower/upper bounds. Hoheisel was the first to prove the existence of a prime in the short interval as above by using the classic zero-density estimates to provide the asymptotic for the Chebyshev psi-function. In 1937, Ingham explicitly described the connection to zero-density estimates used by Hoheisel and improved the value of \theta by providing better zero-density results. The approach of Ingham combined with the zero-density estimates of Huxley (1972) provides us with the distribution of primes in $[x−x^{\theta}, x]$ with $\theta > 7/12$. Further improvement upon the value of \theta was achieved by a different representation of a prime-counting function as the sum of Dirichlet polynomials, namely using sieve methods or Heath-Brown’s identity. We will focus on Harman’s sieve and the weighted zero-density estimates used in the paper of Baker and Harman in 1996. This work provides the best result for $\theta$, achieved by using zero-density estimates. We will discuss the main ideas of the paper by Baker and Harman and simplify some of its parts to show a more explicit connection between zero-density results and the sieved sums, which are used in the paper. This connection should provide a better understanding of which parts should be optimised for further improvements and what the limits of the methods are.