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.
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.
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1294
Keywords: Exponential sums, Riemann zeta function, moments of zeta
Pliego, Javier 1
CC-BY-ND 4.0
@article{JTNB_2024__36_2_725_0,
author = {Pliego, Javier},
title = {Estimates for a three-dimensional exponential sum with monomials},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {725--766},
year = {2024},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {36},
number = {2},
doi = {10.5802/jtnb.1294},
mrnumber = {4830948},
zbl = {07948983},
language = {en},
url = {https://www.numdam.org/articles/10.5802/jtnb.1294/}
}
TY - JOUR AU - Pliego, Javier 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://www.numdam.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 Pliego, Javier %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://www.numdam.org/articles/10.5802/jtnb.1294/ %R 10.5802/jtnb.1294 %G en %F JTNB_2024__36_2_725_0
Pliego, Javier. 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
[1] Decoupling, exponential sums and the Riemann zeta function, J. Am. Math. Soc., Volume 30 (2017) no. 1, pp. 205-224 | Zbl | DOI | MR
[2] Exponential sums with monomials, J. Number Theory, Volume 33 (1989) no. 3, pp. 311-333 | DOI | Zbl | MR
[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 | MR
[5] The fourth power moment of the Riemann zeta function, Proc. Lond. Math. Soc., Volume 38 (1979), pp. 385-422 | DOI | Zbl | MR
[6] Mean-value theorems in the theory of the Riemann zeta-function, Proc. Lond. Math. Soc., Volume 27 (1926), pp. 273-300 | Zbl | MR
[7] The distribution of 4-full numbers, Acta Arith., Volume 67 (1994) no. 2, pp. 165-176 | Zbl | MR
[8] Exponential sums and the abelian group problem, Funct. Approximatio, Comment. Math., Volume 42 (2010) no. 2, pp. 113-129 | Zbl | MR
[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 | MR
[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 | MR
[13] A course of Modern Analysis, Cambridge University Press, 1927, vi+608 pages | MR
[14] Nombres -libres dans les petits intervalles, Acta Arith., Volume 65 (1993) no. 2, pp. 97-116 | Zbl | MR
Cité par Sources :
