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On a problem of Gelfond: the sum of digits of prime numbers. (Sur un problème de Gelfond: la somme des chiffres des nombres premiers.) (English) Zbl 1213.11025
Let \(q\geq 2\) be an integer. Each integer \(n\) can be uniquely written in base \(q\) in the form \[ n=\sum_{k\geq 0}n_kq^k,\quad n_k\in\{0,\ldots,q-1\}. \] Let \(s_q(n)=\sum_{k\geq 0}n_k\) be the sum of digits of \(n\) in the base \(q\). The authors prove:
Theorem 1. For \(q\geq 2\) and \(\alpha\) such that \((q-1)\alpha\in{\mathbb R}\setminus{\mathbb Z}\) there exists \(\sigma_q(\alpha)>0\) such that \[ \sum_{n\leq x}\Lambda(n)e(\alpha s_q(n))=O_{q,\alpha}(x^{1-\sigma_q(\alpha)}), \] where \(\Lambda\) is von Mangoldt’s function and \(e(x)=\exp(2\pi i x)\).
Theorem 2. For \(q\geq 2\) the sequence \((\alpha s_q(p))_{p\in{\mathbb P}}\) is equidistributed modulo \(1\) if and only if \(\alpha\in{\mathbb R}\setminus{\mathbb Q}\). (\({\mathbb P}\) denotes the set of prime numbers.)
Theorem 3. For \(q,m\geq 2\) there exists \(\sigma_{q,m}>0\) such that for every \(a\in{\mathbb Z}\) \[ \text{card}\{p\leq x,s_q(p)\equiv a\bmod m \}=\frac{(m,q-1)}{m}\pi(x;a,(m,q-1))+O_{q,m}(x^{1-\sigma_{q,m}}). \]

MSC:
11A63 Radix representation; digital problems
11A41 Primes
11N37 Asymptotic results on arithmetic functions
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References:
[1] J. Allouche and J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge: Cambridge Univ. Press, 2003. · Zbl 1086.11015 · doi:10.1017/CBO9780511546563
[2] N. L. Bassily and I. Kátai, ”Distribution of the values of \(q\)-additive functions on polynomial sequences,” Acta Math. Hungar., vol. 68, iss. 4, pp. 353-361, 1995. · Zbl 0832.11035 · doi:10.1007/BF01874349
[3] R. Bellman and H. N. Shapiro, ”On a problem in additive number theory,” Ann. of Math., vol. 49, pp. 333-340, 1948. · Zbl 0031.25401 · doi:10.2307/1969281
[4] V. Brun, ”Le crible d’Eratosthène et le théorème de Goldbach,” Christiania Vidensk. Selsk. Skr., vol. 36, 1920. · JFM 47.0162.02
[5] J. R. Chen, ”On the representation of a larger even integer as the sum of a prime and the product of at most two primes,” Sci. Sinica, vol. 16, pp. 157-176, 1973. · Zbl 0319.10056
[6] A. Cobham, ”Uniform tag sequences,” Math. Systems Theory, vol. 6, pp. 164-192, 1972. · Zbl 0253.02029 · doi:10.1007/BF01706087
[7] A. H. Copeland and P. Erdös, ”Note on normal numbers,” Bull. Amer. Math. Soc., vol. 52, pp. 857-860, 1946. · Zbl 0063.00962 · doi:10.1090/S0002-9904-1946-08657-7
[8] J. Coquet, T. Kamae, and M. Mendès France, ”Sur la mesure spectrale de certaines suites arithmétiques,” Bull. Soc. Math. France, vol. 105, iss. 4, pp. 369-384, 1977. · Zbl 0383.10035 · numdam:BSMF_1977__105__369_0 · eudml:87307
[9] C. Dartyge and C. Mauduit, ”Nombres presque premiers dont l’écriture en base \(r\) ne comporte pas certains chiffres,” Journal of Number Theory, vol. 81, iss. 2, pp. 270-291, 2000. · Zbl 0957.11039 · doi:10.1006/jnth.1999.2458
[10] C. Dartyge and C. Mauduit, ”Ensembles de densité nulle contenant des entiers possédant au plus deux facteurs premiers,” J. Number Theory, vol. 91, iss. 2, pp. 230-255, 2001. · Zbl 0988.11042 · doi:10.1006/jnth.2001.2681
[11] C. Dartyge and G. Tenenbaum, ”Sommes des chiffres de multiples d’entiers,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 55, iss. 7, pp. 2423-2474, 2005. · Zbl 1110.11025 · doi:10.5802/aif.2166 · numdam:AIF_2005__55_7_2423_0 · eudml:116259
[12] M. Drmota and J. Rivat, ”The sum-of-digits function of squares,” J. London Math. Soc., vol. 72, iss. 2, pp. 273-292, 2005. · Zbl 1092.11006 · doi:10.1112/S0024610705006769
[13] S. Eilenberg, Automata, Languages, and Machines. Vol. A, New York: Academic Press, 1974, vol. 58. · Zbl 0317.94045
[14] W. J. Ellison, Les Nombres Premiers, Paris: Hermann, 1975. · Zbl 0313.10001
[15] N. Fine, ”The distribution of the sum of digits \(({ mod} p)\),” Bull. Amer. Math. Soc., vol. 71, pp. 651-652, 1965. · Zbl 0148.02005 · doi:10.1090/S0002-9904-1965-11381-7
[16] E. Fouvry and H. Iwaniec, ”Gaussian primes,” Acta Arith., vol. 79, iss. 3, pp. 249-287, 1997. · Zbl 0881.11070 · eudml:206979
[17] E. Fouvry and C. Mauduit, ”Méthodes de crible et fonctions sommes des chiffres,” Acta Arith., vol. 77, iss. 4, pp. 339-351, 1996. · Zbl 0869.11073 · eudml:206923
[18] E. Fouvry and C. Mauduit, ”Sommes des chiffres et nombres presque premiers,” Math. Ann., vol. 305, iss. 3, pp. 571-599, 1996. · Zbl 0858.11050 · doi:10.1007/BF01444238 · eudml:183004
[19] J. Friedlander and H. Iwaniec, ”The polynomial \(X^2+Y^4\) captures its primes,” Ann. of Math., vol. 148, iss. 3, pp. 945-1040, 1998. · Zbl 0926.11068 · doi:10.2307/121034 · www.math.princeton.edu · eudml:120009 · arxiv:math/9811185
[20] J. Friedlander and H. Iwaniec, ”Asymptotic sieve for primes,” Ann. of Math., vol. 148, iss. 3, pp. 1041-1065, 1998. · Zbl 0926.11067 · doi:10.2307/121035 · www.math.princeton.edu · eudml:120008 · arxiv:math/9811186
[21] A. O. Gel’fond, ”Sur les nombres qui ont des propriétés additives et multiplicatives données,” Acta Arith., vol. 13, pp. 259-265, 1967/1968. · Zbl 0155.09003 · eudml:204828
[22] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., New York: Oxford Univ.Press, 1979. · Zbl 0423.10001
[23] G. Harman, ”Primes with preassigned digits,” Acta Arith., vol. 125, iss. 2, pp. 179-185, 2006. · Zbl 1246.11155 · doi:10.4064/aa125-2-4
[24] J. Hartmanis and H. Shank, ”On the recognition of primes by automata,” J. Assoc. Comput. Mach., vol. 15, pp. 382-389, 1968. · Zbl 0164.05201 · doi:10.1145/321466.321470
[25] D. R. Heath-Brown, ”Prime numbers in short intervals and a generalized Vaughan identity,” Canad. J. Math., vol. 34, iss. 6, pp. 1365-1377, 1982. · Zbl 0478.10024 · doi:10.4153/CJM-1982-095-9
[26] D. R. Heath-Brown, ”Primes represented by \(x^3+2y^3\),” Acta Math., vol. 186, iss. 1, pp. 1-84, 2001. · Zbl 1007.11055 · doi:10.1007/BF02392715
[27] E. Heppner, ”Über die Summe der Ziffern natürlicher Zahlen,” Ann. Univ. Sci. Budapest. Eötvös Sect. Math., vol. 19, pp. 41-43 (1977), 1976. · Zbl 0326.10011
[28] G. Hoheisel, ”Primzahlprobleme in der analysis,” Sitz. Preuss. Akad. Wiss., vol. 33, pp. 3-11, 1930. · JFM 56.0172.02
[29] M. N. Huxley, Area, Lattice Points, and Exponential Sums, New York: Oxford Univ. Press, 1996. · Zbl 0861.11002
[30] H. Iwaniec, ”Almost-primes represented by quadratic polynomials,” Invent. Math., vol. 47, iss. 2, pp. 171-188, 1978. · Zbl 0389.10031 · doi:10.1007/BF01578070 · eudml:142575
[31] H. Iwaniec and E. Kowalski, Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2004. · Zbl 1059.11001
[32] M. Keane, ”Generalized Morse sequences,” Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, vol. 10, pp. 335-353, 1968. · Zbl 0162.07201 · doi:10.1007/BF00531855
[33] I. Kátai, ”On the sum of digits of prime numbers,” Ann. Univ. Sci. Budapest. Eötvös Sect. Math., vol. 10, pp. 89-93, 1967. · Zbl 0155.09004
[34] I. Kátai, ”On the sum of digits of primes,” Acta Math. Acad. Sci. Hungar., vol. 30, iss. 1-2, pp. 169-173, 1977. · Zbl 0365.10007 · doi:10.1007/BF01895662
[35] I. Kátai, ”Distribution of digits of primes in \(q\)-ary canonical form,” Acta Math. Hungar., vol. 47, iss. 3-4, pp. 341-359, 1986. · Zbl 0603.10041 · doi:10.1007/BF01953972
[36] I. Kátai and J. Mogyoródi, ”On the distribution of digits,” Publ. Math. Debrecen, vol. 15, pp. 57-68, 1968. · Zbl 0172.06201
[37] G. Lejeune Dirichlet, Mathematische Werke. Bände I, II, Bronx, NY: Chelsea Publ. Co., 1969. · Zbl 0212.00801
[38] 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. Math. Phys. Mass. Inst. Techn., vol. 6, pp. 158-163, 1927. · JFM 53.0265.03
[39] C. Mauduit, ”Propriétés arithmétiques des substitutions,” in Séminaire de Théorie des Nombres, Boston, MA: Birkhäuser, 1992, pp. 177-190. · Zbl 0741.11016
[40] C. Mauduit, ”Multiplicative properties of the Thue-Morse sequence,” Period. Math. Hungar., vol. 43, iss. 1-2, pp. 137-153, 2001. · Zbl 0980.11018 · doi:10.1023/A:1015241900975
[41] C. Mauduit and A. Sárközy, ”On the arithmetic structure of sets characterized by sum of digits properties,” J. Number Theory, vol. 61, iss. 1, pp. 25-38, 1996. · Zbl 0868.11004 · doi:10.1006/jnth.1996.0134
[42] M. Mendès France, ”Les suites à spectre vide et la répartition modulo \(1\),” J. Number Theory, vol. 5, pp. 1-15, 1973. · Zbl 0252.10033 · doi:10.1016/0022-314X(73)90053-X
[43] M. Minsky and S. Papert, ”Unrecognizable sets of numbers,” J. Assoc. Comput. Mach., vol. 13, pp. 281-286, 1966. · Zbl 0166.00601 · doi:10.1145/321328.321337
[44] H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Providence, RI: Published for the Conference Board of the Mathematical Sciences, Washington, DC, Amer. Math. Soc., 1994. · Zbl 0814.11001
[45] M. Olivier, ”Répartition des valeurs de la fonction “somme des chiffres”,” in Séminaire de Théorie des Nombres, 1970-1971 (Univ. Bordeaux I, Talence), Exp. No. 16, Lab. Théorie des Nombres, Centre Nat. Recherche Sci., Talence, 1971, p. 7. · Zbl 0272.10035 · eudml:110764
[46] M. Olivier, ”Sur le développement en base \(g\) des nombres premiers,” C. R. Acad. Sci. Paris Sér. A-B, vol. 272, p. a937-a939, 1971. · Zbl 0215.35801
[47] I. I. Pyateckiui-vShapiro, ”On the distribution of prime numbers in sequences of the form \([f(n)]\),” Mat. Sb., vol. 33(75), pp. 559-566, 1953. · Zbl 0053.02702 · eudml:65750
[48] M. Queffélec, Substitution Dynamical Aystems-Spectral Analysis, New York: Springer-Verlag, 1987. · Zbl 0642.28013
[49] H. Rademacher, ”Beiträge zur Viggo Brunschen methode in der zahlentheorie,” Hamb. Abh., vol. 3, pp. 12-30, 1923. · JFM 49.0128.05
[50] P. Ribenboim, The Book of Prime Number Records, Second ed., New York: Springer-Verlag, 1989. · Zbl 0642.10001
[51] J. Rivat and P. Sargos, ”Nombres premiers de la forme \(\lfloor n^c\rfloor\),” Canad. J. Math., vol. 53, iss. 2, pp. 414-433, 2001. · Zbl 0970.11035 · doi:10.4153/CJM-2001-017-0
[52] A. Rényi, ”Sur la représentation des entiers pairs comme somme d’un nombre premier et d’un nombre presque premier (en russe),” Dokl. Akad. Nauk SSSR, vol. 56, pp. 455-458, 1947.
[53] M. Schützenberger, ”A remark on acceptable sets of numbers,” J. Assoc. Comput. Mach., vol. 15, pp. 300-303, 1968. · Zbl 0165.02204 · doi:10.1145/321450.321461
[54] I. Shiokawa, ”On the sum of digits of prime numbers,” Proc. Japan Acad., vol. 50, pp. 551-554, 1974. · Zbl 0301.10047 · doi:10.3792/pja/1195518832 · projecteuclid.org
[55] W. Sierpiński, ”Sur les nombres premiers ayant des chiffres initiaux et finals donnés,” Acta Arith., vol. 5, pp. 265-266 (1959), 1959. · Zbl 0094.25505 · eudml:206408
[56] G. Tenenbaum, Introduction à la Théorie Analytique et Probabiliste des Nombres, Second ed., Paris: Société Mathématique de France, 1995. · Zbl 0880.11001
[57] R. C. Vaughan, ”An elementary method in prime number theory,” Acta Arith., vol. 37, pp. 111-115, 1980. · Zbl 0448.10037 · eudml:205685
[58] I. M. Vinogradov, The Method of Trigonometrical Sums in the Theory of Numbers, Mineola, NY: Dover Publications, 2004. · Zbl 1093.11001
[59] 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. · JFM 53.0265.02
[60] D. Wolke, ”Primes with preassigned digits,” Acta Arith., vol. 119, iss. 2, pp. 201-209, 2005. · Zbl 1080.11064 · doi:10.4064/aa119-2-5
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