On the distribution of p α modulo one
Journal de théorie des nombres de Bordeaux, Tome 11 (1999) no. 2, pp. 407-423.

Dans cet article, on donne une nouvelle majoration de la discrépance

D(N):=sup 0≤γ≤0 |∑ p≤/N p α ≤γ 1-π(N)γ|
de la suite (p α ), lorsque 5/3≤α>3 et α≠2.

In this paper, we give a new upper-bound for the discrepancy

D(N):=sup 0≤γ≤0 |∑ p≤/N p α ≤γ 1-π(N)γ|
for the sequence (p α ), when 5/3≤α>3 and α≠2.

@article{JTNB_1999__11_2_407_0,
     author = {Cao, Xiaodong and Zhai, Wenguang},
     title = {On the distribution of $p^\alpha $ modulo one},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {407--423},
     publisher = {Universit\'e Bordeaux I},
     volume = {11},
     number = {2},
     year = {1999},
     mrnumber = {1745887},
     zbl = {0988.11027},
     language = {en},
     url = {http://archive.numdam.org/item/JTNB_1999__11_2_407_0/}
}
TY  - JOUR
AU  - Cao, Xiaodong
AU  - Zhai, Wenguang
TI  - On the distribution of $p^\alpha $ modulo one
JO  - Journal de théorie des nombres de Bordeaux
PY  - 1999
SP  - 407
EP  - 423
VL  - 11
IS  - 2
PB  - Université Bordeaux I
UR  - http://archive.numdam.org/item/JTNB_1999__11_2_407_0/
LA  - en
ID  - JTNB_1999__11_2_407_0
ER  - 
%0 Journal Article
%A Cao, Xiaodong
%A Zhai, Wenguang
%T On the distribution of $p^\alpha $ modulo one
%J Journal de théorie des nombres de Bordeaux
%D 1999
%P 407-423
%V 11
%N 2
%I Université Bordeaux I
%U http://archive.numdam.org/item/JTNB_1999__11_2_407_0/
%G en
%F JTNB_1999__11_2_407_0
Cao, Xiaodong; Zhai, Wenguang. On the distribution of $p^\alpha $ modulo one. Journal de théorie des nombres de Bordeaux, Tome 11 (1999) no. 2, pp. 407-423. http://archive.numdam.org/item/JTNB_1999__11_2_407_0/

[1] R C. Baker, G. Kolesnik, On the distribution of pα modulo one, J. Reine Angew. Math. 356 (1985), 174-193. | Zbl

[2] A. Balog, On the fractional part of pθ, Arch. Math. 40 (1983), 434-440. | Zbl

[3] A. Balog, On the distribution of pθ (mod1), Acta. Math. Hungar. 45 (1985), 179-199 | Zbl

[4] E. Bombieri and H. Iwaniec, On the order of ζ(1/2 + it), Ann. Scuola Norm. Sup. Pisa. 13 (1986), 449-472. | Numdam | Zbl

[5] E. Fouvry, H. Iwaniec, Exponential sums with monomials, J. Number Theory 33 (1989), 311-333. | MR | Zbl

[6] S.A. Gritsenko, On a problem of I. M. Vinogradov (in Russian), Mat. Zametki 39 (1986), 625-650 | MR | Zbl

[7] G. Harman, On the distribution of √p modulo one, Mathematika 30 (1983), 104-110. | Zbl

[8] Heath-Brown, D. R. Heath-Brown, The Pjatecskil-Sapiro prime number theory, J. Number Theory 16 (1983), 242-266. | MR | Zbl

[9] R.M. Kaufman, The distribution of √p, Mat. Zametki 26 (1979), 497-504 | Zbl

[10] Leitmann, On the distribution of some sequences, J. London Math. Soc. 14 (1976), 430-432. | MR | Zbl

[11] H.-Q. Liu, On the number of abelian groups of a given order(supplement), Acta Arith. 64 (1993), 285-296. | MR | Zbl

[12] E.C. Titchmarsh, The Theory of Riemann Zeta Function, Oxford, 1951. | MR | Zbl

[13] R.C. Vaughan, The Hardy-Littlewood Method, Cambridge, 1981. | MR | Zbl

[14] I.M. Vinogradov, Special Variants of the Method of Trigonometric sums (in Russian), Izda. Nauka, Moskow, 1976. | MR | Zbl