On Gelfond’s conjecture about the sum of digits of prime numbers

Joël Rivat[1]

  • [1] Institut de Mathématiques de Luminy CNRS-UMR 6206 163 avenue de Luminy Case 907 13288 Marseille Cedex 9, France.

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 2, page 415-422
  • ISSN: 1246-7405

Abstract

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The goal of this paper is to outline the proof of a conjecture of Gelfond [6] (1968) in a recent work in collaboration with Christian Mauduit [11] concerning the sum of digits of prime numbers, reflecting the lecture given in Edinburgh at the Journées Arithmétiques 2007.

How to cite

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Rivat, Joël. "On Gelfond’s conjecture about the sum of digits of prime numbers." Journal de Théorie des Nombres de Bordeaux 21.2 (2009): 415-422. <http://eudml.org/doc/10888>.

@article{Rivat2009,
abstract = {The goal of this paper is to outline the proof of a conjecture of Gelfond [6] (1968) in a recent work in collaboration with Christian Mauduit [11] concerning the sum of digits of prime numbers, reflecting the lecture given in Edinburgh at the Journées Arithmétiques 2007.},
affiliation = {Institut de Mathématiques de Luminy CNRS-UMR 6206 163 avenue de Luminy Case 907 13288 Marseille Cedex 9, France.},
author = {Rivat, Joël},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {2},
pages = {415-422},
publisher = {Université Bordeaux 1},
title = {On Gelfond’s conjecture about the sum of digits of prime numbers},
url = {http://eudml.org/doc/10888},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Rivat, Joël
TI - On Gelfond’s conjecture about the sum of digits of prime numbers
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 2
SP - 415
EP - 422
AB - The goal of this paper is to outline the proof of a conjecture of Gelfond [6] (1968) in a recent work in collaboration with Christian Mauduit [11] concerning the sum of digits of prime numbers, reflecting the lecture given in Edinburgh at the Journées Arithmétiques 2007.
LA - eng
UR - http://eudml.org/doc/10888
ER -

References

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  1. C. Dartyge and G. Tenenbaum, Sommes des chiffres de multiples d’entiers. Ann. Inst. Fourier (Grenoble) 55 (2005), 2423–2474. Zbl1110.11025MR2207389
  2. E. Fouvry and C. Mauduit, Sommes des chiffres et nombres presques premiers. Mathematische Annalen 305 (1996), 571–599. Zbl0858.11050MR1397437
  3. E. Fouvry and C. Mauduit, Méthodes de crible et fonctions sommes des chiffres. Acta Arithmetica 77 (1996), 339–351. Zbl0869.11073MR1414514
  4. J. Friedlander and H. Iwaniec, The polynomial X 2 + Y 4 captures its primes. Ann. of Math. (2) 148 (1998), 945–1040. Zbl0926.11068MR1670065
  5. J. Friedlander and H. Iwaniec, Asymptotic sieve for primes. Ann. of Math. (2) 148 (1998), 1041–1065. Zbl0926.11067MR1670069
  6. A. O. Gelfond, Sur les nombres qui ont des propriétés additives et multiplicatives données. Acta Arithmetica 13 (1968), 259–265. Zbl0155.09003MR220693
  7. G. Harman, Primes with preassigned digits. Acta Arith. 125 (2006), 179–185. Zbl1246.11155MR2277847
  8. D. R. Heath-Brown, Prime numbers in short intervals and a generalized Vaughan identity. Can. J. Math. 34 (1982), 1365–1377. Zbl0478.10024MR678676
  9. D. R. Heath-Brown, Primes represented by x 3 + 2 y 3 . Acta Math. 186 (2001), 1–84. Zbl1007.11055MR1828372
  10. K. Mahler, 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. Zbl53.0265.03
  11. C. Mauduit, J. Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers. Annals of Mathematics, à paraître. 
  12. I. I. Piatetski-Shapiro, On the distribution of prime numbers in sequences of the form [ f ( n ) ] . Mat. Sbornik N.S. 33(75) (1953), 559–566. Zbl0053.02702MR59302
  13. W. Sierpiński, Sur les nombres premiers ayant des chiffres initiaux et finals donnés. Acta Arith. 5 (1959), 265–266. Zbl0094.25505MR109811
  14. R. C. Vaughan, An elementary method in prime number theory. Acta Arithmetica 37 (1980), 111–115. Zbl0448.10037MR598869
  15. I. M. Vinogradov, The method of Trigonometrical Sums in the Theory of Numbers, translated from the Russian, revised and annotated by K.F. Roth and A. Davenport. Interscience, London, 1954. Zbl0055.27504MR2104806
  16. D. Wolke, Primes with preassigned digits. Acta Arith. 119 (2005), 201–209. Zbl1080.11064MR2167722

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