Averages of quadratic twists of long Dirichlet polynomials

TL;DR

This paper studies the average behavior of long Dirichlet polynomials twisted by quadratic characters, confirming predictions from the CFKRS recipe under the Lindel"of Hypothesis for certain lengths.

Contribution

It provides explicit calculations of these averages for polynomial lengths less than the square of the basic scale, verifying the CFKRS recipe's 0- and 1-swap terms.

Findings

Averages computed match CFKRS predictions under Lindel"of Hypothesis.

Verification of 0- and 1-swap terms in the recipe.

Results hold for polynomial lengths less than the square of the scaling parameter.

Abstract

We investigate averages of long Dirichlet polynomials twisted by Kronecker symbols and we compare our result with the recipe of [CFKRS]. We are able to compute these averages in the case that the length of the polynomial is a power less than 2 of the basic scaling parameter on the assumption of the Lindel\"of Hypothesis for $L$-functions of quadratic characters, and we show that the answer is consistent with this recipe. This corresponds, in terms of the recipe, to verifying 0- and 1-swap terms.