Small values of signed harmonic sums

Sandro Bettin; Giuseppe Molteni; Carlo Sanna

doi:10.1016/j.crma.2018.11.007

Number theory

Small values of signed harmonic sums
[Petites valeurs de sommes harmoniques signées]

Présenté par : the Editorial Board

Sandro Bettin ¹ ; Giuseppe Molteni ² ; Carlo Sanna ³

¹ Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy
² Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milano, Italy
³ Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy

Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1062-1074.

Résumé
Abstract

Pour tout $τ \in R$ et tout entier N, soit $m_{N} (τ)$ la distance minimale de τ aux sommes $\sum_{n = 1}^{N} s_{n} / n$ , où $s_{1}, \dots, s_{n} \in {- 1, + 1}$ . On montre que $m_{N} (τ) < \exp (- C {(\log N)}^{2})$ pour tout entier positif N suffisamment grand (dépendant de C et τ), quelle que soit la constante positive C, inférieure à $1 / \log 4$ .

For every $τ \in R$ and every integer N, let $m_{N} (τ)$ be the minimum of the distance of τ from the sums $\sum_{n = 1}^{N} s_{n} / n$ , where $s_{1}, \dots, s_{n} \in {- 1, + 1}$ . We prove that $m_{N} (τ) < \exp (- C {(\log N)}^{2})$ , for all sufficiently large positive integers N (depending on C and τ), where C is any positive constant less than $1 / \log 4$ .

Reçu le : 2018-06-23
Accepté le : 2018-11-08
Publié le : 2018-11-01

DOI : 10.1016/j.crma.2018.11.007

Affiliations des auteurs :

Sandro Bettin ¹ ; Giuseppe Molteni ² ; Carlo Sanna ³

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     title = {Small values of signed harmonic sums},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1062--1074},
     publisher = {Elsevier},
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     number = {11-12},
     year = {2018},
     doi = {10.1016/j.crma.2018.11.007},
     language = {en},
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Sandro Bettin; Giuseppe Molteni; Carlo Sanna. Small values of signed harmonic sums. Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1062-1074. doi : 10.1016/j.crma.2018.11.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.11.007/

Bibliographie
Cité par

[1] S. Bettin; G. Molteni; C. Sanna Greedy approximations by signed harmonic sums and the Thue–Morse sequence, 2018 (preprint) | arXiv

[2] J. Borwein; D. Bailey; R. Girgensohn Experimentation in Mathematics – Computational Paths to Discovery, A K Peters, Ltd., Natick, MA, 2004

[3] D.W. Boyd A p-adic study of the partial sums of the harmonic series, Exp. Math., Volume 3 (1994) no. 4, pp. 287-302

[4] R.E. Crandall Theory of ROOF walks, 2008 http://www.reed.edu/physics/faculty/crandall/papers/ROOF11.pdf

[5] P. Erdős Egy Kürschák-féle elemi számelméleti tétel általánosítása, Mat. Fiz. Lapok, Volume 39 (1932), pp. 17-24

[6] A. Eswarathasan; E. Levine p-Integral harmonic sums, Discrete Math., Volume 91 (1991) no. 3, pp. 249-257

[7] R.K. Guy Unsolved Problems in Number Theory, Problem Books in Mathematics, Springer-Verlag, New York, 2004

[8] K.E. Morrison Random walks with decreasing steps, 1998 https://www.calpoly.edu/~kmorriso/Research/RandomWalks.pdf

[9] K.E. Morrison Cosine products, Fourier transforms, and random sums, Amer. Math. Mon., Volume 102 (1995) no. 8, pp. 716-724 (MR1357488)

[10] C.D. Offner Zeros and growth of entire functions of order 1 and maximal type with an application to the random signs problem, Math. Z., Volume 175 (1980) no. 3, pp. 189-217

[11] The PARI Group http://pari.math.u-bordeaux.fr/ (Bordeaux, PARI/GP, version 2.6.0, 2013, from)

[12] S. Ramanujan Highly composite numbers, Ramanujan J., Volume 1 (1997) no. 2, pp. 119-153 (annotated and with a foreword by Jean-Louis Nicolas and Guy Robin)

[13] J. Riordan Combinatorial Identities, Robert E. Krieger Publishing Co., Huntington, NY, USA, 1979 (reprint of the 1968 original)

[14] J. Sándor; D.S. Mitrinović; B. Crstici Handbook of Number Theory. I, Springer, Dordrecht, The Netherlands, 2006 (second printing of the 1996 original)

[15] C. Sanna On the p-adic valuation of harmonic numbers, J. Number Theory, Volume 166 (2016), pp. 41-46

[16] B. Schmuland Random harmonic series, Amer. Math. Mon., Volume 110 (2003) no. 5, pp. 407-416

[17] E.W. Weisstein Concise Encyclopedia of Mathematics, CRC Press, 2002

[18] J. Wolstenholme On certain properties of prime numbers, Q. J. Pure Appl. Math., Volume 5 (1862), pp. 35-39

[19] B.-L. Wu; Y.-G. Chen On certain properties of harmonic numbers, J. Number Theory, Volume 175 (2017), pp. 66-86

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