On certain sums of number theory

Abstract

We study sums of the shape n x f (⌊x/n⌋) where f is either the von Mangoldt function or the Dirichlet-Piltz divisor functions. We improve previous estimates when f = Λ and f = τ , and provide new results when f = τ r with r 3, breaking the 1 2 -barrier in each case. The functions f = µ 2 and f = 2 ω are also investigated.

On general sums involving the floor function with applications to $k$-free numbers

“…In this paper, we consider the following general result. We can improve previous results for some special arithmetic functions considered by Bordellés [3], Stucky [9] and Liu-Wu-Yang [11].…”

“…has attracted many experts special attention (for example, see [1,3,6,11,12]), where f is a complex-valued arithmetic function and [•] denotes the floor function (i.e. the greatest integer function).…”

“…In fact, for general function f being a positive real-valued arithmetic function, then one can obtain some much better results by involving the theory of Fourier series [10] and exponential sums [5]. For example, one can refer to [3,6,7,8,9]. In this paper, we consider the following general result.…”

On a sum involving certain arithmetic functions and the integral part function

“…Wu [11] and Zhai [12] improved their results independently. Several authors studied the asymptotic formulas for S f when f equals some special arithmetic functions such as τ (n) := the divisor function, (see [1], [2], [6], [7], [8], [9], for instance). With the help of Vaughan identity and the technique of one-dimensional exponential sum, Ma and Wu [8] proved…”

Primes in floor function sets

“…In [8] the authors used exponential sums to find asymptotic bounds and formulas for various classes of arithmetic functions. Subsequent papers by various authors have mainly been focussed on improvements in exponential sums techniques (see [9,11,15,18,19,22,23,24,25,26]).…”