Some explicit formulas for partial sums of Möbius functions
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 273-315.

The purpose of this paper is to give some explicit formulas involving Möbius functions. Such explicit formulas may be known under the generalized Riemann Hypothesis, but unconditional in this paper. Concretely, we prove explicit formulas of partial sums of the Möbius function in arithmetic progressions and partial sums of the Möbius functions on an Abelian number field. In addition, to obtain these explicit formulas, we study a certain finite Euler product appearing from a relation between primitive characters and imprimitive characters.

Le but de cet article est de donner quelques formules explicites faisant intervenir des fonctions de Möbius. De telles formules explicites peuvent être prouvées sous l’hypothèse de Riemann généralisée, mais dans cet article, nous donnons des preuves inconditionnelles. Concrètement, nous prouvons des formules explicites pour des sommes partielles de la fonction de Möbius dans les progressions arithmétiques et pour des sommes partielles des fonctions de Möbius des corps de nombres abéliens. De plus, pour obtenir ces formules explicites, nous étudions un produit eulérien fini provenant d’une relation entre les caractères primitifs et non primitifs.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1162
Classification: 11A25, 11R42
Keywords: Möbius Function, Dirichlet $L$-Functions, Dedekind Zeta-Functions, Gonek–Hejhal Conjecture, Linear Independence Conjecture
Inoue, Shōta 1

1 Department of Mathematics Tokyo Institute of Technology 2-12-1 Ookayama, Meguro-ku Tokyo 152-8551 Japan
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Inoue, Shōta. Some explicit formulas for partial sums of Möbius functions. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 273-315. doi : 10.5802/jtnb.1162. http://archive.numdam.org/articles/10.5802/jtnb.1162/

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