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

top
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

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.