Tauberian Oscillation Theorems and the Distribution of Goldbach numbers

Gautami Bhowmik; Olivier Ramaré; Jan-Christoph Schlage-Puchta

doi:10.5802/jtnb.940

Gautami Bhowmik ¹; Olivier Ramaré ¹; Jan-Christoph Schlage-Puchta ²

¹ Laboratoire Painlevé LABEX-CEMPI Université Lille 1 59655 Villeneuve d’Ascq Cedex FRANCE
² Mathematisches Institut Universität Rostock 18051 Rostock GERMANY

Journal de théorie des nombres de Bordeaux, Volume 28 (2016) no. 2, pp. 291-299.

Abstract
Résumé

Let $D (s) = \sum a_{n} n^{- s}$ be a Dirichlet-series which tends to $\infty$ when approaching some non-real point on the boundary of its domain of convergence from the right. We give a localized oscillation theorem for $\sum_{n \leq x} a_{n}$ , and apply this result to $\sum_{n_{1} + \dots + n_{k} \leq x} Λ (n_{1}) \dots Λ (n_{k})$ .

Soit $D (s) = \sum a_{n} n^{- s}$ une série de Dirichlet qui, en approchant par la droite un point non-réel sur la frontière de son domaine de convergence, tend vers $\infty$ . Nous présentons un théorème d’oscillation localisé pour $\sum_{n \leq x} a_{n}$ , et une application à $\sum_{n_{1} + \dots + n_{k} \leq x} Λ (n_{1}) \dots Λ (n_{k})$ .

Received: 2014-02-07
Accepted: 2015-09-03
Published online: 2017-01-02

MR Zbl

DOI: 10.5802/jtnb.940

Classification: 11M45, 11N37, 11P32
Keywords: Dirichlet series, Omega-results, Goldbach numbers

Author's affiliations:

Gautami Bhowmik ¹; Olivier Ramaré ¹; Jan-Christoph Schlage-Puchta ²

¹ Laboratoire Painlevé LABEX-CEMPI Université Lille 1 59655 Villeneuve d’Ascq Cedex FRANCE
² Mathematisches Institut Universität Rostock 18051 Rostock GERMANY

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     author = {Gautami Bhowmik and Olivier Ramar\'e and Jan-Christoph Schlage-Puchta},
     title = {Tauberian {Oscillation} {Theorems} and the {Distribution} of {Goldbach} numbers},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {291--299},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {28},
     number = {2},
     year = {2016},
     doi = {10.5802/jtnb.940},
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     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.940/}
}

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Gautami Bhowmik; Olivier Ramaré; Jan-Christoph Schlage-Puchta. Tauberian Oscillation Theorems and the Distribution of Goldbach numbers. Journal de théorie des nombres de Bordeaux, Volume 28 (2016) no. 2, pp. 291-299. doi : 10.5802/jtnb.940. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.940/

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