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On the asymptotic behaviour of sums \(\sum g(n)\{x/n\}^ k\). (English) Zbl 0555.10027

Let g(t) be a real-valued, positive, non-decreasing function defined for \(t\geq 1\), then it is the aim of the present paper to study the asymptotic behaviour of sums \(T_ k(x)=\sum_{n\leq x}g(n)\{x/n\}^ k\) where k is an arbitrary positive integer, x is a large real variable and \(\{\). \(\}\) denotes the fractional part. In particular the difference between \(T_ k(x)\) and the integral \(I_ k(x)=\int^{x}_{1}g(t)\{x/t\}^ k dt\) is estimated, yielding the result \[ (1)\quad T_ k(x)=I_ k(x)+O(g(x) x^{1/3} \log x). \] Moreover, it is shown that these differences (for given g and varying k) are all ”approximately equal”, namely that \[ (2)\quad T_ k(x)-I_ k(x)=T_ 1(x)-I_ 1(x)+O(g(x) x^{3/10}). \] Finally it is remarked without proof that the order term in (1) can be refined to O(g(x) \(x^{\theta})\) (with any \(\theta >0,329...)\) and that in (2) to O(g(x) \(x^{109/382})\). As a by-result, the paper provides a fairly general answer to a problem posed in Notices Am. Math. Soc. 29, 150 (1982).

MSC:

11N37 Asymptotic results on arithmetic functions
11L03 Trigonometric and exponential sums (general theory)
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References:

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