On the distribution of p α modulo one

Xiaodong Cao; Wenguang Zhai

Journal de théorie des nombres de Bordeaux (1999)

  • Volume: 11, Issue: 2, page 407-423
  • ISSN: 1246-7405

Abstract

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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 .

How to cite

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Cao, Xiaodong, and Zhai, Wenguang. "On the distribution of $p^\alpha $ modulo one." Journal de théorie des nombres de Bordeaux 11.2 (1999): 407-423. <http://eudml.org/doc/248335>.

@article{Cao1999,
abstract = {In this paper, we give a new upper-bound for the discrepancy\begin\{equation*\}D(N): = \sup \_\{0 \le \gamma \le 0\} | \sum \_\{p \le / N\ \atop \left\lbrace p^\alpha \right\rbrace \le \gamma \} 1-\pi (N) \gamma |\end\{equation*\}for the sequence $(p^\alpha )$, when $5/3 \le \alpha &gt; 3$ and $\alpha \ne 2$.},
author = {Cao, Xiaodong, Zhai, Wenguang},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {upper bound; discrepancy; double large sieve inequality; sums over primes},
language = {eng},
number = {2},
pages = {407-423},
publisher = {Université Bordeaux I},
title = {On the distribution of $p^\alpha $ modulo one},
url = {http://eudml.org/doc/248335},
volume = {11},
year = {1999},
}

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
PB - Université Bordeaux I
VL - 11
IS - 2
SP - 407
EP - 423
AB - In this paper, we give a new upper-bound for the discrepancy\begin{equation*}D(N): = \sup _{0 \le \gamma \le 0} | \sum _{p \le / N\ \atop \left\lbrace p^\alpha \right\rbrace \le \gamma } 1-\pi (N) \gamma |\end{equation*}for the sequence $(p^\alpha )$, when $5/3 \le \alpha &gt; 3$ and $\alpha \ne 2$.
LA - eng
KW - upper bound; discrepancy; double large sieve inequality; sums over primes
UR - http://eudml.org/doc/248335
ER -

References

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  1. [1] R C. Baker, G.Kolesnik, On the distribution of pα modulo one, J. Reine Angew. Math.356 (1985), 174-193. Zbl0546.10027
  2. [2] A. Balog, On the fractional part of pθ, Arch. Math.40 (1983), 434-440. Zbl0517.10038
  3. [3] A. Balog, On the distribution of pθ (mod1), Acta. Math. Hungar.45 (1985), 179-199 Zbl0571.10038
  4. [4] E. Bombieri and H. Iwaniec, On the order of ζ(1/2 + it), Ann. Scuola Norm. Sup. Pisa.13 (1986), 449-472. Zbl0615.10047
  5. [5] E. Fouvry, H. Iwaniec, Exponential sums with monomials, J. Number Theory33 (1989), 311-333. Zbl0687.10028MR1027058
  6. [6] S.A. Gritsenko, On a problem of I. M. Vinogradov (in Russian), Mat. Zametki39 (1986), 625-650 Zbl0612.10029MR850799
  7. [7] G. Harman, On the distribution of √p modulo one, Mathematika30 (1983), 104-110. Zbl0504.10019
  8. [8] Heath-Brown, D. R. Heath-Brown, The Pjatecskil-Sapiro prime number theory, J. Number Theory16 (1983), 242-266. Zbl0513.10042MR698168
  9. [9] R.M. Kaufman, The distribution of √p, Mat. Zametki26 (1979), 497-504 Zbl0417.10030
  10. [10] Leitmann, On the distribution of some sequences, J. London Math. Soc.14 (1976), 430-432. Zbl0343.10025MR432566
  11. [11] H.-Q. Liu, On the number of abelian groups of a given order(supplement), Acta Arith.64 (1993), 285-296. Zbl0790.11074MR1225430
  12. [12] E.C. Titchmarsh, The Theory of Riemann Zeta Function, Oxford, 1951. Zbl0042.07901MR46485
  13. [13] R.C. Vaughan, The Hardy-Littlewood Method, Cambridge, 1981. Zbl0455.10034MR628618
  14. [14] I.M. Vinogradov, Special Variants of the Method of Trigonometric sums (in Russian), Izda. Nauka, Moskow, 1976. Zbl0429.10023MR469878

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