On the fractional parts of x/n and related sequences. II
Annales de l'Institut Fourier, Volume 27 (1977) no. 2, p. 1-30

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)=logn, h(n)=1/logn) 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.

Comme promis dans l’article no I de même titre (Ann. Inst. Fourier, 26-4 (1976), 115-131), nous étudions ici la répartition asymptotique des parties fractionnaires de xh(n)h est une fonction arithmétique (à savoir h(n)=1/n, h(n)=logn, h(n)=1/logn) et n un entier (ou un nombre premier) parcourant l’intervalle [y(x),x)]. On s’est efforcé de démontrer des formes assez fines des théorèmes, encore que certains résultats se prêtent à des améliorations au prix d’une technicité accrue. Des applications arithmétiques seront données plus tard.

@article{AIF_1977__27_2_1_0,
     author = {Saffari, Bahman and Vaughan, R. C.},
     title = {On the fractional parts of $x/n$ and related sequences. II},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {27},
     number = {2},
     year = {1977},
     pages = {1-30},
     doi = {10.5802/aif.649},
     zbl = {0379.10023},
     mrnumber = {58 \#554a},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1977__27_2_1_0}
}
Saffari, Bahman; Vaughan, R. C. On the fractional parts of $x/n$ and related sequences. II. Annales de l'Institut Fourier, Volume 27 (1977) no. 2, pp. 1-30. doi : 10.5802/aif.649. http://www.numdam.org/item/AIF_1977__27_2_1_0/

[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. | MR 13,724e | Zbl 0042.04204

[2] M. N. Huxley, On the difference between consecutive primes, Inventiones Math., 15 (1972), 164-170. | MR 45 #1856 | Zbl 0241.10026

[3] A. E. Ingham, The distribution of prime numbers, Cambridge Tracts in Mathematics and Mathematical Physics, 30, London, 1932. | MR 32 #2391 | Zbl 0006.39701

[4] E. Landau, Vorlesungen uber Zahlentheorie, zweiter Band, Chelsea Pub. Co., New York, 1969.

[5] D. Menchov, Sur les séries de fonctions orthogonales. Première partie. La convergence, Fundamenta Math., 4 (1923), 82-105. | JFM 49.0293.01

[6] C. J. Moreno, The average size of gaps between primes, Mathematika, 21 (1974), 96-100. | MR 53 #7972 | Zbl 0287.10028

[7] K. K. Norton, Numbers with small prime factors, and the least k-th power non-residue, Mem. Am. Math. Soc., 106 (1971). | MR 44 #3948 | Zbl 0211.37801

[8] H. Rademacher, Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen, Math. Ann., 87 (1922), 112-138. | JFM 48.0485.05

[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. | Numdam | MR 56 #2948 | Zbl 0343.10019

[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. | MR 7,48e | Zbl 0063.06869

[11] A. Z. Walfisz, Weylsche Exponentialsummen in der neuren Zahlentheorie, Mathematische Forschungsberichte, XV, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963. | Zbl 0146.06003