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Discrete Uniform Limit Law for a Sum of Additive Functions on Shifted Primes

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Abstract

We consider the limit distribution of a sum of strongly additive arithmetic functions with arguments running

through shifted primes. We obtain sufficient and necessary conditions for the weak convergence of distributions of such sums to the discrete uniform law. We study the case where the functions take values 0 or 1 on primes.

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References

  1. E. Bombieri, On the large sieve, Mathematika, 12(2):201–225, 1965.

    Article  MathSciNet  MATH  Google Scholar 

  2. P.D.T.A. Elliott, Probabilistic Number Theory. I: Mean Value Theorems, Springer, New York, Heidelberg, Berlin, 1979.

    Book  MATH  Google Scholar 

  3. P.D.T.A. Elliott, Probabilistic Number Theory. II: Central Limit Theorems, Springer, New York, Heidelberg, Berlin, 1980.

  4. P.D.T.A. Elliott, Arithmetic Functions and Integer Products, Springer, New York, Berlin, Heidelberg, Tokyo, 1985.

    Book  MATH  Google Scholar 

  5. P.D.T.A. Elliott, The concentration function of additive functions on shifted primes, Acta Math., 173(1):1–35, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  6. P.D.T.A. Elliott, On the Correlation of Multiplicative and the Sum of Additive Arithmetic Functions, Mem.Am. Math. Soc., Vol. 112, No. 538, AMS, Providence, RI, 1994.

  7. A. Hildebrand, An Erdős–Wintner theorem for differences of additive functions, Trans. Am. Math. Soc., 310(1):257–276, 1988.

    MATH  Google Scholar 

  8. I. Kátai, On the distribution of arithmetical functions on the set of primes plus one, Comput. Math., 19(4):278–289, 1968.

    MathSciNet  MATH  Google Scholar 

  9. I. Kátai, On the distribution of arithmetical functions, Acta Math. Acad. Sci. Hung., 20(1–2):69–87, 1969.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Kubilius, Probabilistic Methods in the Theory of Numbers, Transl. Math.Monogr., Vol. 11, AMS, Providence, RI, 1964.

  11. Wm.J. LeVeque, On the size of certain number-theoretic functions, Trans. Am. Math. Soc., 66(2):440–463, 1949.

  12. G. Stepanauskas, The mean values of multiplicative functions on shifted primes, Lith. Math. J., 37(4):443–451, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Stepanauskas, The mean values of multiplicative functions. IV, Publ. Math., 52(3–4):659–681, 1998.

    MathSciNet  MATH  Google Scholar 

  14. G. Stepanauskas and L. Žvinytė, Discrete uniformlimit law for additive functions on shifted primes, Nonlinear Anal. Model. Control, 21(4):437–447, 2016.

    MathSciNet  Google Scholar 

  15. G. Stepanauskas and L. Žvinytė, Discrete uniform distribution for a sum of additive functions, Lith. Math. J., 57(3):391–400, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Šiaulys, Factorial moments of distributions of additive functions, Lith. Math. J., 40(4):389–401, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Šiaulys and G. Stepanauskas, Poisson distribution for a sum of additive functions, Acta Appl.Math., 97(1–3):269–279, 2007.

    MathSciNet  MATH  Google Scholar 

  18. J. Šiaulys and G. Stepanauskas, Poisson distribution for a sum of additive functions on shifted primes, Acta Arith., 130(4):403–414, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Šiaulys and G. Stepanauskas, Some limit laws for strongly additive prime number indicators, Šiauliai Math. Semin., 3(11):235–246, 2008.

    MATH  Google Scholar 

  20. J. Šiaulys and G. Stepanauskas, Binomial limit law for additive prime number indicators, Lith. Math. J., 51(4):562–572, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Šiaulys and G. Stepanauskas, Discrete uniform limit law for additive prime number indicators, in A. Laurinč ikas, E. Manstavičius, and G. Stepanauskas (Eds.), Analytic and Probabilistic Methods in Number Theory, Proceedings of the Fifth International Conference in Honour of J. Kubilius, Palanga, Lithuania, September 25–29, 2011, TEV, Vilnius, 2012, pp. 255–263.

  22. G. Tenenbaum, Introduction to Analysis and Probabilistic Number Theory, Camb. Stud. Adv. Math., Vol. 46, Cambridge Univ. Press, Cambridge, 1995.

  23. N.M. Timofeev, The distribution of values of additive functions on the sequence {p + 1}, Mat. Zametki, 33(6):933–941, 1983.

    MathSciNet  MATH  Google Scholar 

  24. N.M. Timofeev, Arithmetic functions on the set of shifted primes, Proc. Steklov Inst. Math., 207:311–317, 1995.

    MathSciNet  Google Scholar 

  25. N.M. Timofeev and H.H. Usmanov, The distribution of values of a sum of additive functions with shifted arguments, Mat. Zametki, 352(5):113–124, 1992.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Vilnius University, Naugarduko str. 24, LT-03225, Vilnius, Lithuania

    Jonas Šiaulys, Gediminas Stepanauskas & Laura Žvinytė

Authors
  1. Jonas Šiaulys
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  2. Gediminas Stepanauskas
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  3. Laura Žvinytė
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Correspondence to Jonas Šiaulys.

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Šiaulys, J., Stepanauskas, G. & Žvinytė, L. Discrete Uniform Limit Law for a Sum of Additive Functions on Shifted Primes. Lith Math J 58, 235–248 (2018). https://doi.org/10.1007/s10986-018-9389-0

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  • DOI: https://doi.org/10.1007/s10986-018-9389-0

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