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Primes with an average sum of digits

Part of: Multiplicative number theory Exponential sums and character sums Limit theorems Elementary number theory

Published online by Cambridge University Press:  01 March 2009

Michael Drmota ,
Christian Mauduit  and
Joël Rivat
Michael Drmota
Affiliation:
Institute of Discrete Mathematics and Geometry, Technische Universität Wien, Wiedner Hauptstraße 8-10/104, A-1040 Wien, Austria (email: [email protected])
Christian Mauduit
Affiliation:
Institut de Mathématiques de Luminy, CNRS UMR 6206, Université de la Méditerranée, Campus de Luminy, Case 907, 13288 Marseille Cedex 9, France (email: [email protected])
Joël Rivat
Affiliation:
Institut de Mathématiques de Luminy, CNRS UMR 6206, Université de la Méditerranée, Campus de Luminy, Case 907, 13288 Marseille Cedex 9, France (email: [email protected])
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Abstract

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The main goal of this paper is to provide asymptotic expansions for the numbers #{p≤x:p prime,sq(p)=k} for k close to ((q−1)/2)log qx, where sq(n) denotes the q-ary sum-of-digits function. The proof is based on a thorough analysis of exponential sums of the form (where the sum is restricted to p prime), for which we have to extend a recent result by the second two authors.

Keywords

sum-of-digits functionprimesexponential sumscentral limit theorem

MSC classification

Secondary: 11A63: Radix representation; digital problems 11L03: Trigonometric and exponential sums, general 11N05: Distribution of primes 11L20: Sums over primes 60F05: Central limit and other weak theorems
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 2 , March 2009 , pp. 271 - 292
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Bassily, N. L. and Kátai, I., Distribution of the values of q-additive functions on polynomial sequences, Acta Math. Hung. 68 (1995), 353–361.CrossRefGoogle Scholar
[2]Bassily, N. L. and Kátai, I., Distribution of consecutive digits in the q-ary expansion of some subsequences of integers, J. Math. Sci. 78 (1996), 11–17.CrossRefGoogle Scholar
[3]Copeland, A. H. and Erdős, P., Note on normal numbers, Bull. Amer. Math. Soc. 52 (1946), 857–860.CrossRefGoogle Scholar
[4]Coquet, J., Power sums of digital sums, J. Number Theory 22 (1986), 161–176.CrossRefGoogle Scholar
[5]Delange, H., Sur la fonction sommatoire de la fonction “Somme de Chiffres”, Enseignement Math. 21 (1975), 31–77.Google Scholar
[6]Drmota, M. and Rivat, J., The sum-of-digits function of squares, J. London Math. Soc. (2) 72 (2005), 273–292.CrossRefGoogle Scholar
[7]Fouvry, E. and Mauduit, C., Sur les entiers dont la somme des chiffres est moyenne, J. Number Theory 114 (2005), 135–152.CrossRefGoogle Scholar
[8]Grabner, P. J., Kirschenhofer, P., Prodinger, H. and Tichy, R. F., On the moments of the sum-of-digits function, in Applications of Fibonacci numbers, vol. 5 (St. Andrews, 1992). (Kluwer Academic Publishers, Dordrecht, 1993), 263–271.CrossRefGoogle Scholar
[9]Graham, S. and Kolesnik, G., Van der Corput’s method of exponential sums, London Mathematical Society Lecture Note Series, vol. 126 (Cambridge University Press, Cambridge, 1991).CrossRefGoogle Scholar
[10]Heath-Brown, D. R., Prime numbers in short intervals and a generalized Vaughan identity, Canad. J. Math. 34 (1982), 1365–1377.CrossRefGoogle Scholar
[11]Hoheisel, G., Primzahlprobleme in der analysis, Sitz. Preuss. Akad. Wiss. 33 (1930), 3–11.Google Scholar
[12]Iwaniec, H. and Kowalski, E., Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
[13]Katai, I., On the sum of digits of prime numbers, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 10 (1967), 89–93.Google Scholar
[14]Katai, I., On the sum of digits of primes, Acta Math. Acad. Sci. Hungar. 30 (1977), 169–173.CrossRefGoogle Scholar
[15]Katai, I., Distribution of digits of primes in q-ary canonical form, Acta Math. Acad. Sci. Hungar. 47 (1986), 341–359.CrossRefGoogle Scholar
[16]Katai, I. and Mogyorodi, J., On the distribution of digits, Publ. Math. Debrecen 15 (1968), 57–68.CrossRefGoogle Scholar
[17]Mauduit, C. and Rivat, J., Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Ann. Math., to appear.Google Scholar
[18]Mauduit, C. and Rivat, J., La somme des chiffres des carrés, Acta Math., to appear.Google Scholar
[19]Mauduit, C. and Sárközy, A., On the arithmetic structure of the integers whose sum of digits is fixed, Acta Arith. 81 (1997), 145–173.CrossRefGoogle Scholar
[20]Shiokawa, I., On the sum of digits of prime numbers, Proc. Japan Acad. 50 (1974), 551–554.Google Scholar
[21]Stolarsky, K. B., Power and exponential sums of digital sums related to binomial coefficient parity, SIAM J. Appl. Math. 32 (1977), 717–730.CrossRefGoogle Scholar
[22]Vaughan, R. C., An elementary method in prime number theory, Acta Arith. 37 (1980), 111–115.CrossRefGoogle Scholar
[23]Vinogradov, I. M., The method of trigonometrical sums in the theory of numbers (Interscience, London, 1954), translated from the Russian, revised and annotated by K .F. Roth and A. Davenport.Google Scholar