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

Annales de l'institut Fourier (1977)

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

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topSaffari, 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

top- [1] N. G. DE BRUIJN, On the number of positive integers ≤ x and free of prime factors > y, Ned. Akad. Wet. Proc. Ser. A, 54 (1951), 50-60, Indag. Math., 13 (1951), 50-60. Zbl0042.04204MR13,724e
- [2] M. N. HUXLEY, On the difference between consecutive primes, Inventiones Math., 15 (1972), 164-170. Zbl0241.10026MR45 #1856
- [3] A. E. INGHAM, The distribution of prime numbers, Cambridge Tracts in Mathematics and Mathematical Physics, 30, London, 1932. Zbl0006.39701MR32 #2391
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- [5] D. MENCHOV, Sur les séries de fonctions orthogonales. Première partie. La convergence, Fundamenta Math., 4 (1923), 82-105. JFM49.0293.01
- [6] C. J. MORENO, The average size of gaps between primes, Mathematika, 21 (1974), 96-100. Zbl0287.10028MR53 #7972
- [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] H. RADEMACHER, Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen, Math. Ann., 87 (1922), 112-138. JFM48.0485.05
- [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] 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] 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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