Estimates of the Generalized Representation Function in Exponential Sums

Abstract

We study the exponential sums involving the generalized representation function and exponential functions of the form e(αn^β), where 0 ≠ α ∈ ℝ and 0 < β < 1. In particular, an asymptotic formula is proved for the nonlinear exponential sum ${{\sum \nolimits_{{X}< n \leq 2X}}}\, r_{2k}(n)e(\alpha{\sqrt n})$, where r_2k(n) is the generalized representation function.

Keywords

Representation functions / nonlinear exponential sums / Voronoi summation formula / 11L07 / 11L15

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