Sur un problème de Gelfond : la somme des chiffres des nombres premiers

Abstract

In this article we answer a question proposed by Gelfond in 1968. We prove that the sum of digits of prime numbers written in a basis $q\geq 2$ is equidistributed in arithmetic progressions (except for some well known degenerate cases). We prove also that the sequence $(\alpha s_q(p))$ where $p$ runs through the prime numbers is equidistributed modulo $1$ if and only if $\alpha\in \Bbb{R}\setminus\Bbb{Q}$.

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    @book {huxley-1996, MRKEY = {1420620},
      AUTHOR = {Huxley, M. N.},
      TITLE = {Area, Lattice Points, and Exponential Sums},
      SERIES = {London Math. Soc. Monogr.},
      NUMBER = {13},
      PUBLISHER = {Oxford Univ. Press},
      ADDRESS = {New York},
      YEAR = {1996},
      PAGES = {xii+494},
      ISBN = {0-19-853466-3},
      MRCLASS = {11L07 (11J54 11P21)},
      MRNUMBER = {97g:11088},
      MRREVIEWER = {R. C. Baker},
      ZBLNUMBER = {0861.11002},
      }
  • [iwaniec-1978] Go to document H. Iwaniec, "Almost-primes represented by quadratic polynomials," Invent. Math., vol. 47, iss. 2, pp. 171-188, 1978.
    @article {iwaniec-1978, MRKEY = {0485740},
      AUTHOR = {Iwaniec, Henryk},
      TITLE = {Almost-primes represented by quadratic polynomials},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {47},
      YEAR = {1978},
      NUMBER = {2},
      PAGES = {171--188},
      ISSN = {0020-9910},
      MRCLASS = {10H15 (10H30)},
      MRNUMBER = {58 \#5553},
      MRREVIEWER = {J. W. Porter},
      DOI = {10.1007/BF01578070},
      ZBLNUMBER = {0389.10031},
      }
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    @book {iwaniec-kowalski, MRKEY = {2061214},
      AUTHOR = {Iwaniec, Henryk and Kowalski, Emmanuel},
      TITLE = {Analytic Number Theory},
      SERIES = {Amer. Math. Soc. Colloq. Publ.},
      NUMBER = {53},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2004},
      PAGES = {xii+615},
      ISBN = {0-8218-3633-1},
      MRCLASS = {11-02 (11Fxx 11Lxx 11Mxx 11Nxx)},
      MRNUMBER = {2005h:11005},
      MRREVIEWER = {K. Soundararajan},
      ZBLNUMBER = {1059.11001},
      }
  • [keane-1968] Go to document M. Keane, "Generalized Morse sequences," Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, vol. 10, pp. 335-353, 1968.
    @article {keane-1968, MRKEY = {0239047},
      AUTHOR = {Keane, M.},
      TITLE = {Generalized {M}orse sequences},
      JOURNAL = {Z. Wahrscheinlichkeitstheorie und Verw. Gebiete},
      VOLUME = {10},
      YEAR = {1968},
      PAGES = {335--353},
      MRCLASS = {28.70 (47.00)},
      MRNUMBER = {39 \#406},
      MRREVIEWER = {J. C. Oxtoby},
      DOI = {10.1007/BF00531855},
      ZBLNUMBER = {0162.07201},
      }
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    @article {katai-1967, MRKEY = {0241375},
      AUTHOR = {K{á}tai, I.},
      TITLE = {On the sum of digits of prime numbers},
      JOURNAL = {Ann. Univ. Sci. Budapest. Eötvös Sect. Math.},
      FJOURNAL = {Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Mathematica},
      VOLUME = {10},
      YEAR = {1967},
      PAGES = {89--93},
      ISSN = {0524-9007},
      MRCLASS = {10.43},
      MRNUMBER = {39 \#2715},
      MRREVIEWER = {W. Schwarz},
      ZBLNUMBER = {0155.09004},
      }
  • [katai-1977] Go to document I. Kátai, "On the sum of digits of primes," Acta Math. Acad. Sci. Hungar., vol. 30, iss. 1–2, pp. 169-173, 1977.
    @article {katai-1977, MRKEY = {0472747},
      AUTHOR = {K{á}tai, I.},
      TITLE = {On the sum of digits of primes},
      JOURNAL = {Acta Math. Acad. Sci. Hungar.},
      FJOURNAL = {Acta Mathematica Academiae Scientiarum Hungaricae},
      VOLUME = {30},
      YEAR = {1977},
      NUMBER = {1--2},
      PAGES = {169--173},
      ISSN = {0001-5954},
      MRCLASS = {10K20},
      MRNUMBER = {57 \#12437},
      MRREVIEWER = {Ernst Heppner},
      DOI = {10.1007/BF01895662},
      ZBLNUMBER = {0365.10007},
      }
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    @article {katai-1986, MRKEY = {854521},
      AUTHOR = {K{á}tai, I.},
      TITLE = {Distribution of digits of primes in {$q$}-ary canonical form},
      JOURNAL = {Acta Math. Hungar.},
      FJOURNAL = {Acta Mathematica Hungarica},
      VOLUME = {47},
      YEAR = {1986},
      NUMBER = {3-4},
      PAGES = {341--359},
      ISSN = {0236-5294},
      MRCLASS = {11N60},
      MRNUMBER = {88e:11091},
      MRREVIEWER = {V. Sita Ramaiah},
      DOI = {10.1007/BF01953972},
      ZBLNUMBER = {0603.10041},
      }
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    @article {katai-mogyorodi-1968, MRKEY = {0236139},
      AUTHOR = {K{á}tai, I. and Mogyor{ó}di, J.},
      TITLE = {On the distribution of digits},
      JOURNAL = {Publ. Math. Debrecen},
      FJOURNAL = {Publicationes Mathematicae Debrecen},
      VOLUME = {15},
      YEAR = {1968},
      PAGES = {57--68},
      ISSN = {0033-3883},
      MRCLASS = {10.50},
      MRNUMBER = {38 \#4437},
      MRREVIEWER = {W. G. H. Schaal},
      ZBLNUMBER = {0172.06201},
      }
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    @book {dirichlet-1969, MRKEY = {0249268},
      AUTHOR = {Lejeune Dirichlet, G.},
      TITLE = {Mathematische {W}erke. {B}ände {\rm I},
      {\rm II}},
      SERIES = {Herausgegeben auf Veranlassung der Königlich Preussischen Akademie der Wissenschaften von L. Kronecker},
      PUBLISHER = {Chelsea Publ. Co.},
      ADDRESS = {Bronx, NY},
      YEAR = {1969},
      PAGES = {Vol. I: xiv+pp. 3--644; Vol. II: iv+pp. 5--422 (bound as one)},
      MRCLASS = {01.60},
      MRNUMBER = {40 \#2514},
      ZBLNUMBER={0212.00801},
      }
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    @article{mahler-1927,
      author={Mahler, K.},
      TITLE={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},
      JOURNAL={J. Math. Phys. Mass. Inst. Techn.},
      VOLUME={6},
      YEAR={1927},
      PAGES={158--163},
      JFMNUMBER={53.0265.03},
      }
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    @incollection {mauduit-1992, MRKEY = {1476736},
      AUTHOR = {Mauduit, Christian},
      TITLE = {Propriétés arithmétiques des substitutions},
      BOOKTITLE = {Séminaire de {T}héorie des {N}ombres},
      VENUE={ {P}aris, 1989--90},
      SERIES = {Progr. Math.},
      NUMBER = {102},
      PAGES = {177--190},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1992},
      MRCLASS = {11B85 (11K99)},
      MRNUMBER = {98i:11014},
      MRREVIEWER = {Jean-Paul Allouche},
      ZBLNUMBER = {0741.11016},
      }
  • [mauduit-2001] Go to document C. Mauduit, "Multiplicative properties of the Thue-Morse sequence," Period. Math. Hungar., vol. 43, iss. 1-2, pp. 137-153, 2001.
    @article {mauduit-2001, MRKEY = {1830572},
      AUTHOR = {Mauduit, Christian},
      TITLE = {Multiplicative properties of the {T}hue-{M}orse sequence},
      JOURNAL = {Period. Math. Hungar.},
      FJOURNAL = {Periodica Mathematica Hungarica. Journal of the János Bolyai Mathematical Society},
      VOLUME = {43},
      YEAR = {2001},
      NUMBER = {1-2},
      PAGES = {137--153},
      ISSN = {0031-5303},
      CODEN = {PMHGAW},
      MRCLASS = {11K65 (11B85)},
      MRNUMBER = {2002i:11081},
      MRREVIEWER = {Michael Drmota},
      DOI = {10.1023/A:1015241900975},
      ZBLNUMBER = {0980.11018},
      }
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    @article {mauduit-sarkozy-1996, MRKEY = {1418316},
      AUTHOR = {Mauduit, Christian and S{á}rk{ö}zy, Andr{à}s},
      TITLE = {On the arithmetic structure of sets characterized by sum of digits properties},
      JOURNAL = {J. Number Theory},
      FJOURNAL = {Journal of Number Theory},
      VOLUME = {61},
      YEAR = {1996},
      NUMBER = {1},
      PAGES = {25--38},
      ISSN = {0022-314X},
      CODEN = {JNUTA9},
      MRCLASS = {11N60 (11K36 11N56)},
      MRNUMBER = {97g:11107},
      MRREVIEWER = {Yuri Bilu},
      DOI = {10.1006/jnth.1996.0134},
      ZBLNUMBER = {0868.11004},
      }
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    @article {mendes-france-1973, MRKEY = {0319909},
      AUTHOR = {Mend{è}s France, Michel},
      TITLE = {Les suites à spectre vide et la répartition modulo {$1$}},
      JOURNAL = {J. Number Theory},
      FJOURNAL = {Journal of Number Theory},
      VOLUME = {5},
      YEAR = {1973},
      PAGES = {1--15},
      ISSN = {0022-314X},
      MRCLASS = {10F40},
      MRNUMBER = {47 \#8450},
      MRREVIEWER = {L. Kuipers},
      DOI = {10.1016/0022-314X(73)90053-X},
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      }
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    @article {minsky-papert-1966, MRKEY = {0207481},
      AUTHOR = {Minsky, Marvin and Papert, Seymour},
      TITLE = {Unrecognizable sets of numbers},
      JOURNAL = {J. Assoc. Comput. Mach.},
      FJOURNAL = {Journal of the Association for Computing Machinery},
      VOLUME = {13},
      YEAR = {1966},
      PAGES = {281--286},
      ISSN = {0004-5411},
      MRCLASS = {94.40},
      MRNUMBER = {34 \#7296},
      MRREVIEWER = {M. Yoeli},
      DOI = {10.1145/321328.321337},
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    @book {montgomery-1994, MRKEY = {1297543},
      AUTHOR = {Montgomery, Hugh L.},
      TITLE = {Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis},
      SERIES = {{\rm CBMS} Regional Conf. Ser. Math.},
      NUMBER = {84},
      PUBLISHER = {Published for the Conference Board of the Mathematical Sciences, Washington, DC, Amer. Math. Soc.},
      ADDRESS={Providence, RI},
      YEAR = {1994},
      PAGES = {xiv+220},
      ISBN = {0-8218-0737-4},
      MRCLASS = {11-02 (11Kxx 11L07 11Mxx 11Nxx)},
      MRNUMBER = {96i:11002},
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      ZBLNUMBER = {0814.11001},
      }
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    @incollection {olivier-1971, MRKEY = {0392889},
      AUTHOR = {Olivier, Michel},
      TITLE = {Répartition des valeurs de la fonction ``somme des chiffres''},
      BOOKTITLE = {Séminaire de {T}héorie des {N}ombres, 1970--1971 ({U}niv. {B}ordeaux {I},
      {T}alence), {\rm {E}xp. {N}o. 16}},
      PAGES = {7},
      PUBLISHER = {Lab. Théorie des Nombres, Centre Nat. Recherche Sci., Talence},
      YEAR = {1971},
      MRCLASS = {10K10 (10H25)},
      MRNUMBER = {52 \#13702},
      MRREVIEWER = {Jean-Marie De Koninck},
      ZBLNUMBER={0272.10035},
      }
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    @article {olivier-1971-CRAS, MRKEY = {0277492},
      AUTHOR = {Olivier, Michel},
      TITLE = {Sur le développement en base {$g$} des nombres premiers},
      JOURNAL = {C. R. Acad. Sci. Paris Sér. {\rm A-B}},
      VOLUME = {272},
      YEAR = {1971},
      PAGES = {A937--A939},
      MRCLASS = {10.43},
      MRNUMBER = {43 \#3225},
      MRREVIEWER = {P. Erd{ő}s},
      ZBLNUMBER={0215.35801},
      }
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    @article {piatetski-shapiro-1953, MRKEY = {0059302},
      AUTHOR = {Pyatecki{\u\i}-{Š}hapiro, I. I.},
      TITLE = {On the distribution of prime numbers in sequences of the form {$[f(n)]$}},
      JOURNAL = {Mat. Sb.},
      VOLUME = {33(75)},
      YEAR = {1953},
      PAGES = {559--566},
      MRCLASS = {10.0X},
      MRNUMBER = {15,507e},
      MRREVIEWER = {R. Bellman},
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      }
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    @book {queffelec-1987, MRKEY = {924156},
      AUTHOR = {Queff{é}lec, Martine},
      TITLE = {Substitution Dynamical Aystems---Spectral Analysis},
      SERIES = {Lecture Notes in Mathematics},
      NUMBER = {1294},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1987},
      PAGES = {xiv+240},
      ISBN = {3-540-18692-1},
      MRCLASS = {54H20 (11K06 28D05 58F19)},
      MRNUMBER = {89g:54094},
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    @article{rademacher-1924,
      author={Rademacher, H.},
      TITLE={Beiträge zur Viggo Brunschen methode in der zahlentheorie},
      JOURNAL={Hamb. Abh.},
      VOLUME={3},
      YEAR={1923},
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      }
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    @book {ribenboim-1989, MRKEY = {1016815},
      AUTHOR = {Ribenboim, Paulo},
      TITLE = {The Book of Prime Number Records},
      EDITION = {{S}econd},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1989},
      PAGES = {xxiv+479},
      ISBN = {0-387-97042-8},
      MRCLASS = {11N05 (01A99 11A41 11Y11)},
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      }
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    @article {rivat-sargos-2001, MRKEY = {1820915},
      AUTHOR = {Rivat, Jo{ë}l and Sargos, Patrick},
      TITLE = {Nombres premiers de la forme {$\lfloor n\sp c\rfloor$}},
      JOURNAL = {Canad. J. Math.},
      FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques},
      VOLUME = {53},
      YEAR = {2001},
      NUMBER = {2},
      PAGES = {414--433},
      ISSN = {0008-414X},
      CODEN = {CJMAAB},
      MRCLASS = {11N05 (11L07)},
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    @article{renyi-1947,
      author={Rényi, A.},
      TITLE={Sur la représentation des entiers pairs comme somme d'un nombre premier et d'un nombre presque premier (en russe)},
      JOURNAL={Dokl. Akad. Nauk {\rm SSSR}},
      VOLUME={56},
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    @article {schutzenberger-1968, MRKEY = {0238634},
      AUTHOR = {Sch{ü}tzenberger, Marcel-Paul},
      TITLE = {A remark on acceptable sets of numbers},
      JOURNAL = {J. Assoc. Comput. Mach.},
      FJOURNAL = {Journal of the Association for Computing Machinery},
      VOLUME = {15},
      YEAR = {1968},
      PAGES = {300--303},
      ISSN = {0004-5411},
      MRCLASS = {94.50},
      MRNUMBER = {38 \#6910},
      MRREVIEWER = {S. Huzino},
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      ZBLNUMBER={0165.02204},
      }
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    @article {shiokawa-1974-2, MRKEY = {0369238},
      AUTHOR = {Shiokawa, Iekata},
      TITLE = {On the sum of digits of prime numbers},
      JOURNAL = {Proc. Japan Acad.},
      FJOURNAL = {Proceedings of the Japan Academy},
      VOLUME = {50},
      YEAR = {1974},
      PAGES = {551--554},
      ISSN = {0021-4280},
      MRCLASS = {10A30},
      MRNUMBER = {51 \#5473},
      MRREVIEWER = {I. Katai},
      URL = {http://projecteuclid.org/euclid.pja/1195518832},
      ZBLNUMBER = {0301.10047},
      }
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      AUTHOR = {Sierpi{ń}ski, W.},
      TITLE = {Sur les nombres premiers ayant des chiffres initiaux et finals donnés},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
      VOLUME = {5},
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      ISSN = {0065-1036},
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      AUTHOR = {Tenenbaum, G{é}rald},
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      NUMBER = {1},
      EDITION = {{S}econd},
      PUBLISHER = {Société Mathématique de France},
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      PAGES = {xv+457},
      ISBN = {2-85629-032-9},
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      AUTHOR = {Vaughan, R. C.},
      TITLE = {An elementary method in prime number theory},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
      VOLUME = {37},
      YEAR = {1980},
      PAGES = {111--115},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {10H20},
      MRNUMBER = {82c:10055},
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      AUTHOR = {Vinogradov, I. M.},
      TITLE = {The Method of Trigonometrical Sums in the Theory of Numbers},
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      YEAR = {2004},
      PAGES = {x+180},
      ISBN = {0-486-43878-3},
      MRCLASS = {11Lxx (01A75)},
      MRNUMBER = {2005f:11172},
      ZBLNUMBER = {1093.11001},
      }
  • [wiener-1927] N. Wiener, "The spectrum of an array and its application to the study of the translation properties of a simple class of arithmetical functions. I: The spectrum of an array," J. Math. Phys. Mass. Inst. Techn., vol. 6, pp. 145-157, 1927.
    @article{wiener-1927,
      author={Wiener, N.},
      TITLE={The spectrum of an array and its application to the study of the translation properties of a simple class of arithmetical functions. I: The spectrum of an array},
      JOURNAL={J. Math. Phys. Mass. Inst. Techn.},
      VOLUME={6},
      YEAR={1927},
      PAGES={145--157},
      JFMNUMBER={53.0265.02},
      }
  • [wolke-2005] Go to document D. Wolke, "Primes with preassigned digits," Acta Arith., vol. 119, iss. 2, pp. 201-209, 2005.
    @article {wolke-2005, MRKEY = {2167722},
      AUTHOR = {Wolke, Dieter},
      TITLE = {Primes with preassigned digits},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Acta Arithmetica},
      VOLUME = {119},
      YEAR = {2005},
      NUMBER = {2},
      PAGES = {201--209},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {11N05},
      MRNUMBER = {2006e:11142},
      MRREVIEWER = {Filip Saidak},
      DOI = {10.4064/aa119-2-5},
      ZBLNUMBER = {1080.11064},
      }

Authors

Christian Mauduit

Institut de Mathématiques de Luminy
Campus de Luminy, Case 907
13288 Marseille Cedex 9
France

Joël Rivat

Institut de Mathématiques de Luminy
Campus de Luminy, Case 907
13288 Marseille Cedex 9
France