## hugh L Montgomery - The combinatorics of moment calculations

hrj:168 - Hardy-Ramanujan Journal, January 1, 2010, Volume 33 - 2010 - https://doi.org/10.46298/hrj.2010.168
The combinatorics of moment calculations

Authors: hugh L Montgomery

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.

Volume: Volume 33 - 2010
Published on: January 1, 2010
Imported on: March 3, 2015
Keywords: moments of binomial random variables, reduced residues,rimes in short intervals,[MATH] Mathematics [math]