Acta Mathematica

La somme des chiffres des carrés

Christian Mauduit and Joël Rivat

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Résumé

L’objet de cet article est de répondre à une question posée par Gelfond en 1968 en montrant que la somme des chiffres des carrés écrits en base q ⩾ 2 est équirépartie dans les progressions arithmétiques.

Abstract

In this article we answer a question proposed by Gelfond in 1968. We prove that the sum of digits of squares written in a basis q ⩾ 2 is equidistributed in arithmetic progressions.

Article information

Source
Acta Math. Volume 203, Number 1 (2009), 107-148.

Dates
Received: 29 August 2007
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892410

Digital Object Identifier
doi:10.1007/s11511-009-0040-0

Mathematical Reviews number (MathSciNet)
MR2545827

Zentralblatt MATH identifier
1278.11076

Subjects
Primary: 11A63: Radix representation; digital problems {For metric results, see 11K16} 11L03: Trigonometric and exponential sums, general 11N05: Distribution of primes
Secondary: 11L20: Sums over primes 11N60: Distribution functions associated with additive and positive multiplicative functions

Rights
2009 © Institut Mittag-Leffler

Citation

Mauduit, Christian; Rivat, Joël. La somme des chiffres des carrés. Acta Math. 203 (2009), no. 1, 107--148. doi:10.1007/s11511-009-0040-0. https://projecteuclid.org/euclid.acta/1485892410


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References

  • A llouche, J. P. & S hallit, J., Automatic Sequences. Cambridge University Press, Cambridge, 2003.
  • B assily, N. L. & K átai, I., Distribution of the values of q-additive functions on polynomial sequences. Acta Math. Hungar., 68 (1995), 353–361.
  • B ellman, R. & S hapiro, H. N., On a problem in additive number theory. Ann. of Math., 49 (1948), 333–340.
  • B üchi, J. R., Weak second-order arithmetic and finite automata. Z. Math. Logik Grundlag. Math., 6 (1960), 66–92.
  • C obham, A., Uniform tag sequences. Math. Systems Theory, 6 (1972), 164–192.
  • C oquet, J., K amae, T. & M endès F rance, M., Sur la mesure spectrale de certaines suites arithmétiques. Bull. Soc. Math. France, 105 (1977), 369–384.
  • D artyge, C. & T enenbaum, G., Sommes des chiffres de multiples d’entiers. Ann. Inst. Fourier (Grenoble), 55:7 (2005), 2423–2474.
  • — Congruences de sommes de chiffres de valeurs polynomiales. Bull. London Math. Soc., 38 (2006), 61–69.
  • D avenport, H. & E rdõs, P., Note on normal decimals. Canadian J. Math., 4 (1952), 58–63.
  • D rmota, M. & R ivat, J., The sum-of-digits function of squares. J. London Math. Soc., 72 (2005), 273–292.
  • F ogg, N. P., Substitutions in Dynamics, Arithmetics and Combinatorics. Lecture Notes in Mathematics, 1794. Springer, Berlin–Heidelberg, 2002.
  • G el’fond, A. O., Sur les nombres qui ont des propriétés additives et multiplicatives données. Acta Arith., 13 (1967/1968), 259–265.
  • G raham, S. W. & K olesnik, G., van der Corput’s method of exponential sums. London Mathematical Society Lecture Note Series, 126. Cambridge University Press, Cambridge, 1991.
  • K eane, M., Generalized Morse sequences. Z. Wahrsch. und Verw. Gebiete, 10 (1968), 335–353.
  • M ahler, K., The spectrum of an array and its application to the study of the translation properties of a simple class of arithmetical functions. II: on the translation properties of a simple class of arithmetical functions. J. of Math. Phys. Mass. Inst. Techn., 6 (1927), 158–163.
  • M auduit, C. & R ivat, J., Répartition des fonctions q-multiplicatives dans la suite ([nc])nN, c>1. Acta Arith., 71 (1995), 171–179.
  • — J., Propriétés q-multiplicatives de la suite [nc], c>1. Acta Arith., 118 (2005), 187–203.
  • — Sur un problème de Gelfond: la somme des chiffres des nombres premiers. À paraître dans Ann. of Math.
  • M endès F rance, M., Les suites à spectre vide et la répartition modulo 1. J. Number Theory, 5 (1973), 1–15.
  • M insky, M. & P apert, S., Unrecognizable sets of numbers. J. Assoc. Comput. Mach., 13 (1966), 281–286.
  • M ontgomery, H. L., Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis. CBMS Regional Conference Series in Mathematics, 84. Conference Board of the Mathematical Sciences, Washington, DC, 1994.
  • N iederreiter, H. & S hparlinski, I. E., On the distribution of inversive congruential pseudorandom numbers in parts of the period. Math. Comp., 70 (2001), 1569–1574.
  • P eter, M., The summatory function of the sum-of-digits function on polynomial sequences. Acta Arith., 104 (2002), 85–96.
  • P iatetski-S hapiro, I. I., On the distribution of prime numbers in sequences of the form [f(n)]. Mat. Sb., 33:75 (1953), 559–566 (en russe).
  • P rouhet, E., Mémoire sur quelques relations entre les puissances des nombres. C. R. Acad. Sc. Paris, 33 (1851), 225.
  • Q ueffélec, M., Substitution Dynamical Systems–Spectral Analysis. Lecture Notes in Mathematics, 1294. Springer, Berlin–Heidelberg, 1987.
  • Ritchie, R. W., Finite automata and the set of squares. J. Assoc. Comput. Mach., 10 (1963), 528–531.
  • S ierpiński, W., Elementary Theory of Numbers. North-Holland Mathematical Library, 31. North-Holland, Amsterdam, 1988.
  • W iener, N., The spectrum of an array and its applications to the study of the translation properties of a simple class of arithmetical functions. I: The spectrum of an array. J. of Math. Phys. Mass. Inst. Techn., 6 (1927), 145–157.