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

• Volume: 27, Issue: 2, page 1-30
• ISSN: 0373-0956

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## Abstract

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As promised in the first paper of this series (Ann. Inst. Fourier, 26-4 (1976), 115-131), these two articles deal with the asymptotic distribution of the fractional parts of $xh\left(x\right)$ where $h$ is an arithmetical function (namely $h\left(n\right)=1/n$, $h\left(n\right)=logn$, $h\left(n\right)=1/logn$) and $n$ is an integer (or a prime order) running over the interval $\left[y\left(x\right),x\right)\right]$. The results obtained are rather sharp, although one can improve on some of them at the cost of increased technicality. Number-theoretic applications will be given later on.

## How to cite

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Saffari, Bahman, and Vaughan, R. C.. "On the fractional parts of $x/n$ and related sequences. II." Annales de l'institut Fourier 27.2 (1977): 1-30. <http://eudml.org/doc/74318>.

@article{Saffari1977,
abstract = {As promised in the first paper of this series (Ann. Inst. Fourier, 26-4 (1976), 115-131), these two articles deal with the asymptotic distribution of the fractional parts of $xh(x)$ where $h$ is an arithmetical function (namely $h(n)=1/n$, $h(n)=\log n$, $h(n)=1/\log n$) and $n$ is an integer (or a prime order) running over the interval $[y(x),x)]$. The results obtained are rather sharp, although one can improve on some of them at the cost of increased technicality. Number-theoretic applications will be given later on.},
author = {Saffari, Bahman, Vaughan, R. C.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {2},
pages = {1-30},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the fractional parts of $x/n$ and related sequences. II},
url = {http://eudml.org/doc/74318},
volume = {27},
year = {1977},
}

TY - JOUR
AU - Saffari, Bahman
AU - Vaughan, R. C.
TI - On the fractional parts of $x/n$ and related sequences. II
JO - Annales de l'institut Fourier
PY - 1977
PB - Association des Annales de l'Institut Fourier
VL - 27
IS - 2
SP - 1
EP - 30
AB - As promised in the first paper of this series (Ann. Inst. Fourier, 26-4 (1976), 115-131), these two articles deal with the asymptotic distribution of the fractional parts of $xh(x)$ where $h$ is an arithmetical function (namely $h(n)=1/n$, $h(n)=\log n$, $h(n)=1/\log n$) and $n$ is an integer (or a prime order) running over the interval $[y(x),x)]$. The results obtained are rather sharp, although one can improve on some of them at the cost of increased technicality. Number-theoretic applications will be given later on.
LA - eng
UR - http://eudml.org/doc/74318
ER -

## References

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2. [2] M. N. HUXLEY, On the difference between consecutive primes, Inventiones Math., 15 (1972), 164-170. Zbl0241.10026MR45 #1856
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4. [4] E. LANDAU, Vorlesungen uber Zahlentheorie, zweiter Band, Chelsea Pub. Co., New York, 1969.
5. [5] D. MENCHOV, Sur les séries de fonctions orthogonales. Première partie. La convergence, Fundamenta Math., 4 (1923), 82-105. JFM49.0293.01
6. [6] C. J. MORENO, The average size of gaps between primes, Mathematika, 21 (1974), 96-100. Zbl0287.10028MR53 #7972
7. [7] K. K. NORTON, Numbers with small prime factors, and the least k-th power non-residue, Mem. Am. Math. Soc., 106 (1971). Zbl0211.37801MR44 #3948
8. [8] H. RADEMACHER, Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen, Math. Ann., 87 (1922), 112-138. JFM48.0485.05
9. [9] B. SAFFARI and R.-C. VAUGHAN, On the fractional parts of x/n and related sequences. I, Annales de l'Institut Fourier, 26,4 (1976), 115-131. Zbl0343.10019MR56 #2948
10. [10] A. SELBERG, On the normal density of primes in small intervals, and the difference between consecutive primes, Arch. Math. Naturvid., 47 (1943), 87-105. Zbl0063.06869MR7,48e
11. [11] A. Z. WALFISZ, Weylsche Exponentialsummen in der neuren Zahlentheorie, Mathematische Forschungsberichte, XV, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963. Zbl0146.06003

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