L-functions in Analytic Number Theory: Winston Heap

We discuss the role of long Dirichlet polynomials in number theory. We first survey some applications of mean values of long Dirichlet polynomials over primes in the theory of the Riemann zeta function which includes central limit theorems and pair correlation of zeros. We then give some examples showing how, on assuming the Riemann Hypothesis, one can compute asymptotics for such mean values without using the Hardy-Littlewood conjectures for additive correlations of the von-Mangoldt functions.