no. 2
On Gelfond’s conjecture about the sum of digits of prime numbers
Rivat, Joël1
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, Volume 21 (2009) no. 2, pp. 415-422.

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.

Dans cet article nous exposons les étapes importantes de la preuve de la conjecture de Gelfond [6] (1968) dans un travail récent en collaboration avec Christian Mauduit [11] concernant la somme des chiffres des nombres premiers, dans l’esprit de l’exposé donné à Edimbourg dans le cadre des Journées Arithmétiques 2007.

DOI: 10.5802/jtnb.678
Rivat, Joël 1

1 Institut de Mathématiques de Luminy CNRS-UMR 6206 163 avenue de Luminy Case 907 13288 Marseille Cedex 9, France. 
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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, Volume 21 (2009) no. 2, pp. 415-422. doi : 10.5802/jtnb.678. http://archive.numdam.org/articles/10.5802/jtnb.678/

[1] C. Dartyge and G. Tenenbaum, Sommes des chiffres de multiples d’entiers. Ann. Inst. Fourier (Grenoble) 55 (2005), 2423–2474. | Numdam | MR | Zbl

[2] E. Fouvry and C. Mauduit, Sommes des chiffres et nombres presques premiers. Mathematische Annalen 305 (1996), 571–599. | MR | Zbl

[3] E. Fouvry and C. Mauduit, Méthodes de crible et fonctions sommes des chiffres. Acta Arithmetica 77 (1996), 339–351. | MR | Zbl

[4] J. Friedlander and H. Iwaniec, The polynomial X 2 +Y 4 captures its primes. Ann. of Math. (2) 148 (1998), 945–1040. | MR | Zbl

[5] J. Friedlander and H. Iwaniec, Asymptotic sieve for primes. Ann. of Math. (2) 148 (1998), 1041–1065. | MR | Zbl

[6] A. O. Gelfond, Sur les nombres qui ont des propriétés additives et multiplicatives données. Acta Arithmetica 13 (1968), 259–265. | MR | Zbl

[7] G. Harman, Primes with preassigned digits. Acta Arith. 125 (2006), 179–185. | MR

[8] D. R. Heath-Brown, Prime numbers in short intervals and a generalized Vaughan identity. Can. J. Math. 34 (1982), 1365–1377. | MR | Zbl

[9] D. R. Heath-Brown, Primes represented by x 3 +2y 3 . Acta Math. 186 (2001), 1–84. | MR | Zbl

[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.

[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. | MR | Zbl

[13] W. Sierpiński, Sur les nombres premiers ayant des chiffres initiaux et finals donnés. Acta Arith. 5 (1959), 265–266. | MR | Zbl

[14] R. C. Vaughan, An elementary method in prime number theory. Acta Arithmetica 37 (1980), 111–115. | MR | Zbl

[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. | MR | Zbl

[16] D. Wolke, Primes with preassigned digits. Acta Arith. 119 (2005), 201–209. | MR | Zbl

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