Nair-Tenenbaum bounds uniform with respect to the discriminant

TL;DR

This paper refines bounds on sums of arithmetic functions over polynomial values by explicitly incorporating the discriminant, improving the understanding of how the discriminant influences these sums in number theory.

Contribution

It provides an explicit version of Nair and Tenenbaum's bound with an optimal dependence on the polynomial's discriminant, enhancing previous results.

Findings

Derived an explicit bound with optimal discriminant dependence

Extended Nair and Tenenbaum's general bound to explicit form

Improved understanding of discriminant's role in sum estimates

Abstract

A common problem in analytic number theory is to bound the sum of an arithmetic function over a set of integers. Nair and Tenenbaum found a very general bound that applies to short sums of a multivariable arithmetic function over polynomial values, under certain standard conditions on the growth of that function. Their bound features an implicit dependency on the discriminant of the relevant polynomial. In our paper we obtain an analogous bound with an explicit dependency on the discriminant, which is optimal in the discriminant aspect.