Some explicit formulas for partial sums of Möbius functions

Inoue, Shōta

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 2, pp. 273-315.

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.

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.

Classification : 11A25, 11R42
Mots clés : Möbius Function, Dirichlet $L$-Functions, Dedekind Zeta-Functions, Gonek–Hejhal Conjecture, Linear Independence Conjecture

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

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

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

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

[6] Gonek, Steven M. 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] Hejhal, Dennis A. On the distribution of $log|{\zeta }^{\prime }(\frac{1}{2}+it)|$, Number theory, trace formulas, and discrete groups, Academic Press Inc., 1989, pp. 343-370 | Zbl

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

[9] Kühn, Patrick; Robles, Nicolas; Roy, Arindam 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] Montgomery, Hugh L. Zeros of $L$-Functions, Invent. Math., Volume 8 (1969), pp. 346-354 | DOI | Zbl

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

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

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

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

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

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

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

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