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A CLASS OF ARITHMETIC FUNCTIONS ON PSL2(Z)

  • 투고 : 2011.12.04
  • 발행 : 2013.03.31

초록

In [3] and [2], Atanassov introduced the two arithmetic functions $$I(n)=\prod_{p^{\alpha}{\parallel}n}\;p^{1/{\alpha}}\;and\;R(n)=\prod_{p^{\alpha}{\parallel}n}\;p^{{\alpha}-1}$$ called the irrational factor and the restrictive factor, respectively. Alkan, Ledoan, Panaitopol, and the authors explore properties of these arithmetic functions in [1], [7], [8] and [9]. In the present paper, we generalize these functions to a larger class of elements of $PSL_2(\mathbb{Z})$, and explore some of the properties of these maps.

키워드

참고문헌

  1. E. Alkan, A. H. Ledoan, and A. Zaharescu, Asymptotic behavior of the irrational factor, Acta Math. Hungar. 121 (2008), no. 3, 293-305. https://doi.org/10.1007/s10474-008-7212-9
  2. K. T. Atanassov, Restrictive factor: definition, properties and problems, Notes Number Theory Discrete Math. 8 (2002), no. 4, 117-119.
  3. K. T. Atanassov, Irrational factor: definition, properties and problems, Notes Number Theory Discrete Math. 2 (1996), no. 3, 42-44.
  4. F. Boca and R. Gologan, On the distribution of the free path length of the linear flow in a honeycomb, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 3, 1043-1075. https://doi.org/10.5802/aif.2457
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  7. A. Ledoan and A. Zaharescu, Real moments of the restrictive factor, Proc. Indian Acad. Sci. Math. Sci. 119 (2009), no. 4, 559-566. https://doi.org/10.1007/s12044-009-0039-7
  8. L. Panaitopol, Properties of the Atanassov functions, Adv. Stud. Contemp. Math. (Kyungshang) 8 (2004), no. 1, 55-58.
  9. P. Spiegelhalter and A. Zaharescu, Strong and weak Atanassov pairs, Proc. Jangjeon Math. Soc. 14 (2011), no. 3, 355-361.
  10. E. C. Titchmarsh, The Theory of Functions, Oxford University Press, London, 1939.
  11. E. C. Titchmarsh, The theory of the Riemann zeta-function, Edited and with a preface by D. R. Heath-Brown, The Clarendon Press Oxford University Press, New York, 1986.
  12. I. M. Vinogradov, A new estimate of the function ${\zeta}$(1+it), Izv. Akad. Nauk SSSR. Ser. Mat. 22 (1958), 161-164.
  13. A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, VEB Deutsche Verlag der Wissenschaften, Berlin, 1963.

피인용 문헌

  1. Analytic continuation and asymptotics of Dirichlet series with partitions vol.433, pp.1, 2016, https://doi.org/10.1016/j.jmaa.2015.07.044