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

Bahman Saffari; R. C. Vaughan

Annales de l'institut Fourier (1977)

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

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 x h ( 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.

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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  1. [1] N. G. DE BRUIJN, On the number of positive integers ≤ x and free of prime factors &gt; y, Ned. Akad. Wet. Proc. Ser. A, 54 (1951), 50-60, Indag. Math., 13 (1951), 50-60. Zbl0042.04204MR13,724e
  2. [2] M. N. HUXLEY, On the difference between consecutive primes, Inventiones Math., 15 (1972), 164-170. Zbl0241.10026MR45 #1856
  3. [3] A. E. INGHAM, The distribution of prime numbers, Cambridge Tracts in Mathematics and Mathematical Physics, 30, London, 1932. Zbl0006.39701MR32 #2391
  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

Citations in EuDML Documents

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  1. Alessandro Zaccagnini, A conditional density theorem for the zeros of the Riemann zeta-function
  2. Claus Bauer, On the sum of a prime and the kth power of a prime
  3. Alessandro Languasco, Alberto Perelli, On Linnik's theorem on Goldbach numbers in short intervals and related problems
  4. Karin Halupczok, Goldbach’s problem with primes in arithmetic progressions and in short intervals
  5. Paolo Codecà, On the properties of oscillation and almost periodicity of certain convolutions
  6. T. Zhan, A generalization of the Goldbach-Vinogradov theorem
  7. Alberto Perelli, János Pintz, On the exceptional set for the 2 k -twin primes problem
  8. Chao Hua Jia, On the exceptional set of Goldbach numbers in a short interval
  9. Alessandro Zaccagnini, Primes in almost all short intervals

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