Exponential Sums by Irrationality Exponent

2025/04/09 by Cha, Byungchul, Kim, Dong Han
#11L07 (Primary) #37E10 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.06726

Abstract

In this article, we give an asymptotic bound for the exponential sum of the Möbius function ∑_{n ≤ x} μ(n) e(αn) for a fixed irrational number α∈ℝ. This exponential sum was originally studied by Davenport and he obtained an asymptotic bound of x(log x)^-A for any A≥0. Our bound depends on the irrationality exponent η of α. If η≤ 5/2, we obtain a bound of x^{4/5 + ε} and, when η≥ 5/2, our bound is x^{(2η-1)/2η+ ε}. This result extends a result of Murty and Sankaranarayanan, who obtained the same bound in the case η= 2.

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