Averages of Long Dirichlet Polynomials

Based on the work of Sandro Bettin and Brian Conrey, our paper extends the methods that were originally used for moments of the zeta function and variance of the k-fold divisor function to the logarithmic derivative of zeta. It uses the correspondence with Random Matrix Theory, where the characteristic polynomials of unitary matrices have served as a powerful analogue for the Riemann zeta function. We can define the Riemann zeta function as

\zeta(s) = \sum_{n=0}^{\infty} \frac{1}{n^{s}}, \qquad \Re{s} > 1

which is a Dirichlet series. We can also express the k-fold product of the Riemann zeta as another Dirichlet series:

\zeta(s)^k = \sum_{n=0}^{\infty} \frac{d_k(n)}{n^s}, \qquad \Re{s} > 1.

The truncated moments for the above Dirichlet series were studied by Bettin and Conrey in their paper “Averages of Long Dirichlet Polynomials”. In our paper, we look at the truncated moments for the logarithmic derivative, defined as

\left( \frac{\zeta’}{\zeta}(s) \right)^k = \sum_{n=0}^{\infty}\frac{\Lambda^{*k}}{n^s}, \qquad \Re{s>1. }

The coefficients of interest to us are convolutions of the von Mangoldt function, not the generalised von Mangoldt which is also commonly studied. This paper along with two others, will focus on extending results that are known for moments of zeta to the moments of the logarithmmic derivative.

Access the preprint on the “Publications and preprints” page.

Averages of Long Dirichlet Polynomials

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