Wednesday, 28 June 2023


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.



Valeriia Starichkova 

Research area

Number Theory


UNSW Canberra


Wednesday 28 June 2023, 3.00 pm


4082 (Anita B. Lawrence Center)