The exponential sum over squarefree integers

Abstract

We show that the set of prime numbers has exponential alternating complexity, proving a conjecture by Fijalkow. We further show that the set of squarefree integers has essentially maximal possible alternating complexity.

MOMENT ESTIMATES FOR EXPONENTIAL SUMS OVER k-FREE NUMBERS

“…Their results were refined by Mirksy [6] and recent work on this topic can be found in [2], for example. As another immediate application we mention that the pointwise bounds for S k (α) in [7] and [8] can be combined with our L p results to improve on the minor-arc estimate in Theorem 1.3 of [1].…”