Nair–Tenenbaum uniform with respect to the discriminant–ERRATUM

Abstract

In this note we wish to correct several mistakes which appeared in our paper [1]. Fortunately, all the results claimed in that reference may be recovered, with small modifications to the statements and the argument. We are grateful to Nathan Ng for drawing our attention to an issue in the application of Theorem 6 to divisor sums, and to Régis de la Bretèche for helpful discussions on the problem addressed here.Congruence conditions. We use the notation as well as page, equation and theorem numbers from [1]. The most problematic part of our paper is the lower bound in Theorem 6, which is incorrect as stated, and as a consequence the upper bounds in Theorem 5 and Corollaries 1-2 are not sharp as claimed (although they are still valid). We explain here how to recover a lower bound on the sum considered in Theorem 6, together with a matching upper bound in Theorems 5 and 6, by modifying the function ρ R (a 1 , . . . , a r ) which encodes certain polynomial congruences.The source of the error in Theorem 6 is at the end of p. 423, where the last displayed equation is wrong: the conditionsε 3 alone are insufficient to guarantee that the a i are the unique integers such that, but we may well have p| ji a j , preventing us from concluding that p > x ε 3 . To address this we introduce the following property, dependent on integers a 1 , . . . , a r , n:The workaround we find is to replace the function ρ R defined in (2·8), in all instances where it appears, by the functionρ R defined bywe explain shortly why this is possible. First observe that, by the Chinese remainder theorem,ρ R is also multiplicative, and sinceρ R ρ R it also satisfies the upper bound (2·9). Thereforeρ R behaves as ρ R for analytic purposes, and it remains to show that the substitution can be performed. In our paper, the function ρ R arises exclusively through applications available at https://www.cambridge.org/core/terms. https://doi

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