Nous calculons une collection de moments mixtes de la fonction zêta de Riemann à l’aide d’une équation fonctionnelle approximative pour le produit des fonctions zêta. Comme application, nous fournissons des estimations pour des sommes exponentielles tridimensionnelles à phase monomiale qui sont dans certains cas plus précises que celles provenant des approches nécessitant l’utilisation des estimations existantes des sommes analogues.
We compute a collection of mixed moments of the Riemann-zeta function by means of an approximate functional equation for the product of zeta functions. As an application we provide estimates for three-dimensional exponential sums with monomials which are in some instances sharper than those stemming from approaches entailing the use of existing bounds pertaining to analogous sums.
Accepté le :
Publié le :
Mots clés : Exponential sums, Riemann zeta function, moments of zeta
CC-BY-ND 4.0
@article{JTNB_2024__36_2_725_0,
author = {Javier Pliego},
title = {Estimates for a three-dimensional exponential sum with monomials},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {725--766},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {36},
number = {2},
year = {2024},
doi = {10.5802/jtnb.1294},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1294/}
}
TY - JOUR AU - Javier Pliego TI - Estimates for a three-dimensional exponential sum with monomials JO - Journal de théorie des nombres de Bordeaux PY - 2024 SP - 725 EP - 766 VL - 36 IS - 2 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1294/ DO - 10.5802/jtnb.1294 LA - en ID - JTNB_2024__36_2_725_0 ER -
%0 Journal Article %A Javier Pliego %T Estimates for a three-dimensional exponential sum with monomials %J Journal de théorie des nombres de Bordeaux %D 2024 %P 725-766 %V 36 %N 2 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1294/ %R 10.5802/jtnb.1294 %G en %F JTNB_2024__36_2_725_0
Javier Pliego. Estimates for a three-dimensional exponential sum with monomials. Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 2, pp. 725-766. doi : 10.5802/jtnb.1294. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1294/
[1] Decoupling, exponential sums and the Riemann zeta function, J. Am. Math. Soc., Volume 30 (2017) no. 1, pp. 205-224 | DOI | Zbl
[2] Exponential sums with monomials, J. Number Theory, Volume 33 (1989) no. 3, pp. 311-333 | DOI | Zbl
[3] Van der Corput’s method for exponential sums, London Mathematical Society Lecture Note Series, 126, London Mathematical Society, 1991, 120 pages | DOI
[4] Contributions to the theory of the Riemann zeta function and the theory of the distribution of primes, Acta Math., Volume 41 (1918), pp. 119-196 | DOI | Zbl
[5] The fourth power moment of the Riemann zeta function, Proc. Lond. Math. Soc., Volume 38 (1979), pp. 385-422 | DOI | Zbl
[6] Mean-value theorems in the theory of the Riemann zeta-function, Proc. Lond. Math. Soc., Volume 27 (1926), pp. 273-300 | Zbl
[7] The distribution of 4-full numbers, Acta Arith., Volume 67 (1994) no. 2, pp. 165-176 | Zbl
[8] Exponential sums and the abelian group problem, Funct. Approximatio, Comment. Math., Volume 42 (2010) no. 2, pp. 113-129 | Zbl
[9] Mixed moments of the Riemann zeta function (2022) | arXiv
[10] Three-dimensional exponential sums with monomials, J. Reine Angew. Math., Volume 591 (2006), pp. 1-20 | DOI | Zbl
[11] Multiple exponential sums with monomials and their applications in number theory, Acta Math., Volume 87 (2000) no. 4, pp. 333-354 | Zbl
[12] The theory of the Riemann zeta-function, Oxford Science Publications, Clarendon Press, 1986, x+412 pages
[13] A course of Modern Analysis, Cambridge University Press, 1927, vi+608 pages
[14] Nombres -libres dans les petits intervalles, Acta Arith., Volume 65 (1993) no. 2, pp. 97-116 | Zbl
Cité par Sources :