Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T22:36:13.120Z Has data issue: false hasContentIssue false
  • English
  • Français

On the convergence of Diophantine Dirichlet series

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Diophantine approximation, transcendental number theory Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  23 February 2012

T. Rivoal
T. Rivoal
Affiliation:
Institut Fourier, CNRS UMR 5582, Université Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint-Martin d'Hères cedex, France ([email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study various Dirichlet series of the form ∑n≥1f(πnα)/ns, where α is an irrational number and f(x) is a trigonometric function like cot(x), 1/sin(x) or 1/sin2(x). The convergence is slow and strongly depends on the Diophantine properties of α. We provide necessary and sufficient convergence conditions using the continued fraction of α. We also show that any one of our series is equal to a related series, which converges much faster, defined in term of iterations of the continued fraction operator α↦{1/α}.

Keywords

Diophantine approximationDirichlet seriescontinued fractions

MSC classification

Secondary: 11J70: Continued fractions and generalizations 11M41: Other Dirichlet series and zeta functions 11K70: Harmonic analysis and almost periodicity
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 2 , June 2012 , pp. 513 - 541
Copyright
Copyright © Edinburgh Mathematical Society 2012

References

1.Adams, W. W., Asymptotic Diophantine approximations to e, Proc. Natl Acad. Sci. USA 55 (1966), 28–31.CrossRefGoogle ScholarPubMed
2.Báez-Duarte, L., Balazard, M., Landreau, B. and Saias, E., Étude de l'autocorrélation multiplicative de la fonction ‘partie fractionnaire’, Ramanujan J. 9(2) (2005), 215–240.CrossRefGoogle Scholar
3.Balazard, M. and Martin, B., Comportement local moyen de la fonction de Brjuno, preprint (2010).Google Scholar
4.Beck, J., Randomness of the square root of 2 and the giant leap, Part 1, Period. Math. Hungar. 60(2) (2010), 137–242.CrossRefGoogle Scholar
5.Berndt, B., Modular transformations and generalizations of several formulae of Ramanujan, Rocky Mt. J. Math. 7 (1977), 147–189.CrossRefGoogle Scholar
6.Bernstein, L., The Jacobi–Perron algorithm, its theory and applications, Lecture Notes in Mathematics, Volume 207 (Springer, 1971).CrossRefGoogle Scholar
7.Chowla, S. D., Some problems of Diophantine approximation, I, Math. Z. 33 (1931), 544–563.CrossRefGoogle Scholar
8.Davenport, H., On some infinite series involving arithmetical functions, I, Q. J. Math. 8 (1937), 8–13.CrossRefGoogle Scholar
9.Davenport, H., On some infinite series involving arithmetical functions, II, Q. J. Math. 8 (1937), 313–320.CrossRefGoogle Scholar
10.Davenport, H., Note on an identity connected with Diophantine approximation, Proc. Camb. Phil. Soc. 34 (1938), 27.CrossRefGoogle Scholar
11.de la Bretèche, R. and Tenenbaum, G., Séries trigonométriques à coefficients arithmétiques, J. Analyse Math. 92 (2004), 1–79.CrossRefGoogle Scholar
12.Grosswald, E., Comments on some formulae of Ramanujan, Acta Arith. 21 (1972), 25–34.CrossRefGoogle Scholar
13.Hardy, G. H. and Littlewood, J. E., Some problems of Diophantine approximation, I, The fractional part of nkΘ, Acta Math. 37 (1914), 155–191.CrossRefGoogle Scholar
14.Hardy, G. H. and Littlewood, J. E., Some problems of Diophantine approximation, II, The trigonometrical series associated with the elliptic ϑ-functions, Acta Math. 37 (1914), 193–239.CrossRefGoogle Scholar
15.Hardy, G. H. and Littlewood, J. E., Some problems of Diophantine approximation: the analytic character of the sum of a Dirichlet's series considered by Hecke, Hamb. Math. Abh. 3 (1923), 57–68.CrossRefGoogle Scholar
16.Hardy, G. H. and Littlewood, J. E., Some problems of Diophantine approximation: a series of cosecant, Bull. Calcutta Math. Soc. 20 (1930), 251–266.Google Scholar
17.Khintchine, A. Ya., Continued fractions (Dover, New York, 1997).Google Scholar
18.Kruse, A. H., Estimates of k−s 〈kx〉−t, Acta Arith. 12 (1966), 229–261.CrossRefGoogle Scholar
19.Laczkovich, M., Equidecomposability and discrepancy: a solution of Tarski's circlesquaring problem, J. Reine Angew. Math. 404 (1990), 77–117.Google Scholar
20.Marcovecchio, R., The Rhin–Viola method for log(2), Acta Arith. 139(2) (2009), 147–184.CrossRefGoogle Scholar
21.Martin, B., Nouvelles identités de Davenport, Funct. Approx. Comment. Math. 37 (2007), 293–327.CrossRefGoogle Scholar
22.Nishioka, K., Mahler functions and transcendence, Lecture Notes in Mathematics, Volume 1631 (Springer, 1996).CrossRefGoogle Scholar
23.Murty, M. Ram, Smyth, C. and Wang, R. J., Zeros of Ramanujan polynomials, J. Ramanujan Math. Soc. 26 (2011), 107–125.Google Scholar
24.Roth, K. F., Rational approximations to algebraic numbers, Mathematika 2 (1955), 1–20 (see also corrigendum, p. 168).CrossRefGoogle Scholar
25.Salikhov, V. Kh., On the irrationality measure of π, Russ. Math. Surv. 63 (2008), 570–572.CrossRefGoogle Scholar
26.Schoissengeier, J., On the convergence of a series of Bundschuh, Uniform Distribution Theory 2 (2007), 107–113.Google Scholar
27.Schoissengeier, J. and Tričković, S. B., On the divergence of a certain series, J. Math. Analysis Applic. 324 (2006), 238–247.CrossRefGoogle Scholar
28.Tenenbaum, G., Introduction à la théorie analytique et probabiliste des nombres, 3rd edition (Belin, Paris, 2008).Google Scholar
29.Walfisz, A., Über einige trigonometrische Summen, Math. Z. 33 (1931), 564–601.CrossRefGoogle Scholar
30.Wilton, J. R., An approximate functional equation with applications to a problem of Diophantine approximation, J. Reine Angew. Math. 169 (1933), 219–237.CrossRefGoogle Scholar