The combinatorics of moment calculations

Hugh L Montgomery (1)

1 Department of Mathematics - University of Michigan

In this paper, we consider the distribution of primes in intervals of length $\asymp\log x$; this leads to insights concerning Poisson random variables. Later, we consider reduced residues $({\rm mod}\,q)$ in short intervals, which similarly motivates us to derive further information concerning moments of binomial random variables. This discussion is extended further, where we consider the distribution of primes in intervals of length $\asymp x^{\theta}$ where $0<\theta<1$. Our analysis gives rise to a family of polynomials.

Mots clés: reduced residues; rimes in short intervals; moments of binomial random variables

Domaines: Mathématiques [math]

Hugh L Montgomery. The combinatorics of moment calculations. Hardy-Ramanujan Journal, 2010, Volume 33 - 2010, pp.2 - 22. ⟨10.46298/hrj.2010.168⟩. ⟨hal-01112382⟩