On the fractional parts of $x/n$ and related sequences. I

On the fractional parts of $x/n$ and related sequences. II

“…We now show that n this is not always so, even under fairly reasonable conditions. In particular, the following theorem justifies our remark below Theorem 4 of [9] to the effect that taking ^ a^{y) in n that theorem can lose a factor as large as log log y. …”

“…-As x -> oo, (1.9) ©,,(a) = S ,7^--^ + 0(^ log ^)- If y is quite close to x, the error term in (1.11) is not very good, and at first sight one might hope to do better. However, on inspecting F(oc, ^) one finds that the error can indeed be this large, and is essentially due to the irregular behaviour of F(a, ^) as a function of S at the points 2,3, ... (see Lemma 4 of [9]). …”

“…It is a simple application of Theorem 5 of [9], that if 0 < 6 < 1, y = x^ and ^,,y(a) has a limit as x tends to infinity, then the limit must be a. Moreover, this can be sharpened along the lines of Theorem 2.…”

“…In this paper we assume the notation of [9]. Throughout, the implicit constants in the 0, <^ and > notations are absolute unless otherwise indicated.…”

“…We recall the definition of F(a, ^, a) (ct. [9], (2.4), (2.5)). The connection between @^y and the Dirichlet divisor problem can be seen, for example, via By adapting the Van der Corput method of trigonometric sums it would be possible to improve the error term here, much as in the Dirichlet divisor problem.…”