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On the mean value of the function \( {\bar{S}_k}(n) \)

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An asymptotic formula is constructed for the mean value of the function \( {\bar{S}_k}(n) \) dual to the Smarandache function S k (n). The O- and Ω-estimates for the second moment of the remainder are obtained.

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  1. 1.

    J. Balacenoin and V. Seleacu, “History of the Smarandache function,” Smarandache Not. J., 10, No. 1–3, 192–201 (1999).

  2. 2.

    Liping Ding, “On the mean value of the Smarandache ceil function,” Sci. Magna., 1, No. 2, 74–77 (2005).

  3. 3.

    Yaming Lu, “On a dual function of the Smarandache function,” in: Research on Smarandache Problems in the Number heory, Hexis, Phoenix, AZ), 2 (2005), pp. 55–57.

  4. 4.

    H. Montgomery and R. Vaughan, “Hilbert’s inequality,” J. London Math. Soc., 2, No. 8, 73–82 (1974).

  5. 5.

    A. Ivić, The Riemann Zeta-Function, Wiley, New York (1985).

  6. 6.

    K. Ramachandra, “On the mean-value and Omega-theorems for the Riemann zeta-function,” Tata Inst. Found. Res., Bombay (1995).

  7. 7.

    Xiaoyan Li, “The mean value of the kth Smarandache dual function,” in: Proc. of the Fifth Internat. Conf. on the Number Theory and Smarandache Notions, Hexis (2009), pp. 128–132.

  8. 8.

    A. Ivić, “On the number of subgroups of finite abelian groups,” J. Théorie Nombres Bordeaux, 9, 371–381 (1997).

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Author information

Correspondence to P.D. Varbanets.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 4, pp. 448–458, April, 2011

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Varbanets, P., Kirbat, S.A. On the mean value of the function \( {\bar{S}_k}(n) \) . Ukr Math J 63, 516 (2011). https://doi.org/10.1007/s11253-011-0520-1

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  • Asymptotic Formula
  • Dirichlet Series
  • Schwarz Inequality
  • Riemann Zeta Function
  • Euler Gamma Function