Lower bounds of discrete moments of the derivatives of the Riemann zeta-function on the critical line

Christ, Thomas; Kalpokas, Justas

doi:10.5802/jtnb.836

Christ, Thomas ¹ ; Kalpokas, Justas ²

¹ Department of Mathematics Würzburg University Emil-Fischer-Str. 40, 97074 Würzburg
² Faculty of Mathematics and Informatics Vilnius University Naugarduko 24, 03225 Vilnius, Lithuania

Journal de théorie des nombres de Bordeaux, Tome 25 (2013) no. 2, pp. 285-305.

Résumé
Abstract

Nous établissons des bornes inférieures inconditionnelles pour certains moments discrets de la fonction zêta de Riemann et de ses dérivées dans la bande critique. Nous utilisons ces moments discrets pour donner des bornes inférieures inconditionnelles pour les moments continus $I_{k, l} (T) = \int_{0}^{T} {| ζ^{(l)} (\frac{1}{2} + i t) |}^{2 k} d t$ , où $l$ est un entier positif et $k \geq 1$ un nombre rationnel. En particulier, ces bornes inférieures sont de l’ordre de grandeur attendu pour $I_{k, l} (T)$ .

We establish unconditional lower bounds for certain discrete moments of the Riemann zeta-function and its derivatives on the critical line. We use these discrete moments to give unconditional lower bounds for the continuous moments $I_{k, l} (T) = \int_{0}^{T} {| ζ^{(l)} (\frac{1}{2} + i t) |}^{2 k} d t$ , where $l$ is a non-negative integer and $k \geq 1$ a rational number. In particular, these lower bounds are of the expected order of magnitude for $I_{k, l} (T)$ .

EuDML MR Zbl

DOI : 10.5802/jtnb.836

Affiliations des auteurs :

Christ, Thomas ¹ ; Kalpokas, Justas ²

¹ Department of Mathematics Würzburg University Emil-Fischer-Str. 40, 97074 Würzburg
² Faculty of Mathematics and Informatics Vilnius University Naugarduko 24, 03225 Vilnius, Lithuania

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     author = {Christ, Thomas and Kalpokas, Justas},
     title = {Lower bounds of discrete moments of the derivatives of the {Riemann} zeta-function on the critical line},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {285--305},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {25},
     number = {2},
     year = {2013},
     doi = {10.5802/jtnb.836},
     zbl = {1288.11082},
     mrnumber = {3228308},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/jtnb.836/}
}

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Christ, Thomas; Kalpokas, Justas. Lower bounds of discrete moments of the derivatives of the Riemann zeta-function on the critical line. Journal de théorie des nombres de Bordeaux, Tome 25 (2013) no. 2, pp. 285-305. doi : 10.5802/jtnb.836. https://www.numdam.org/articles/10.5802/jtnb.836/

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