On some sums involving the integral part function

Abstract

Denote by τ k (n), ω(n) and µ 2 (n) the number of representations of n as product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [t] be the integral part of real number t. For f = ω, 2 ω , µ 2 , τ k , we prove that10k−1 and ε > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordellès.

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

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In this paper, we consider sums related the floor function. We can improve previous results for some special arithmetic functions considered by Bordellés [3], Stucky [9] and Liu-Wu-Yang [6]. We can also give a refined result for previous results for general function considered by Bordellés, Dai, Heyman, Pan and Shparlinski [1] and Wu [11].

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“…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]).…”