Some explicit formulas for partial sums of Möbius functions
Shōta Inoue1
1 Department of Mathematics Tokyo Institute of Technology 2-12-1 Ookayama, Meguro-ku Tokyo 152-8551 Japan
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
Mots-clés : Möbius Function, Dirichlet $L$-Functions, Dedekind Zeta-Functions, Gonek–Hejhal Conjecture, Linear Independence Conjecture

Shōta Inoue 1

1 Department of Mathematics Tokyo Institute of Technology 2-12-1 Ookayama, Meguro-ku Tokyo 152-8551 Japan
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Shōta Inoue. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1162/

[1] Krystyna M. Bartz On some complex explicit formulae connected with the Möbius function. II, Acta Arith., Volume 57 (1991) no. 4, pp. 295-305 | DOI | MR | Zbl

[2] Gautami Bhowmik; Karin Halupczok; Kohji Matsumoto; Yuta Suzuki Goldbach representations in arithmetic progressions and zeros of Dirichlet L-functions, Mathematika, Volume 65 (2019) no. 1, pp. 57-97 | DOI | MR | Zbl

[3] J. Brian Conrey The Riemann hypothesis, Notices Am. Math. Soc., Volume 50 (2003) no. 3, pp. 341-353 | MR | Zbl

[4] Moubariz Z. Garaev; Ayyadurai Sankaranarayanan The sum involving derivative of ζ(s) over simple zeros, J. Number Theory, Volume 117 (2006) no. 1, pp. 122-130 | DOI | MR

[5] Steven M. Gonek On negative moments of the Riemann zeta-function, Mathematika, Volume 36 (1989) no. 1, pp. 71-88 | DOI | MR | Zbl

[6] Steven M. Gonek An explicit formula of Landau and its applications to the theory of the zeta-function, A tribute to Emil Grosswald: number theory and related analysis (Contemporary Mathematics), Volume 143, Springer, 1993, pp. 395-413 | MR | Zbl

[7] Dennis A. Hejhal On the distribution of log|ζ ′ (1 2+it)|, Number theory, trace formulas, and discrete groups, Academic Press Inc., 1989, pp. 343-370 | Zbl

[8] Albert E. Ingham On two conjectures in the theory of numbers, Am. J. Math., Volume 64 (1942), pp. 313-319 | DOI | MR | Zbl

[9] Patrick Kühn; Nicolas Robles; Arindam Roy On a class of functions that satisfies explicit formulae involving the Möbius function, Ramanujan J., Volume 38 (2015) no. 2, pp. 383-422 | DOI | Zbl

[10] Hugh L. Montgomery Zeros of L-Functions, Invent. Math., Volume 8 (1969), pp. 346-354 | DOI | Zbl

[11] Hugh L. Montgomery Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, 84, American Mathematical Society, 1994 | MR | Zbl

[12] Hugh L. Montgomery; Robert C. Vaughan Multiplicative Number Theory I. Classical Theory, Cambridge Studies in Advanced Mathematics, 97, Cambridge University Press, 2007 | MR | Zbl

[13] M. Ram Murty; Jeanine Van Order Counting integral ideals in a number field, Expo. Math., Volume 25 (2007) no. 1, pp. 53-66 | DOI | MR | Zbl

[14] Władysław Narkiewicz Elementary and Analytic Theory of Algebraic Numbers, Springer, 1990 (substantially revised and extended) | Zbl

[15] Nathan Ng The distribution of the summatory function of the Möbius function, Proc. Lond. Math. Soc., Volume 89 (2004), pp. 361-389 | MR | Zbl

[16] Kanakanahalli Ramachandra; Ayyadurai Sankaranarayanan Notes on the Riemann zeta-function, J. Indian Math. Soc., New Ser., Volume 57 (1991) no. 1-4, pp. 67-77 | MR

[17] Kannan Soundararajan Partial sums of the Möbius function, J. Reine Angew. Math., Volume 631 (2009), pp. 141-152

[18] Edward C. Titchmarsh The theory of the Riemann zeta-Function, Oxford University Press, 1986 (revised and with a preface by D. R. Heath-Brown)

[19] Lynnelle Ye Bounding sums of the Möbius function over arithmetic progressions (2014) (https://arxiv.org/abs/1406.7326, to appear in Acta. Arith.)

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