대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제50권2호
  • /
  • Pages.601-610
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  • 2013
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  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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A CLASS OF ARITHMETIC FUNCTIONS ON PSL2(Z)

  • 투고 : 2011.12.04
  • 발행 : 2013.03.31
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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.

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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
  5. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Sixth, Revised by D. R. Heath-Brown and J. H. Silverman, With a foreword by Andrew Wiles, Oxford University Press, Oxford, 2008.
  6. N. M. Korobov, Estimates of trigonometric sums and their applications, Uspehi Mat. Nauk 13 (1958), no. 4, 185-192.
  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