Research article Special Issues

The inverse of tails of Riemann zeta function, Hurwitz zeta function and Dirichlet L-function

  • Received: 20 March 2024 Revised: 27 April 2024 Accepted: 30 April 2024 Published: 13 May 2024
  • MSC : 11B83, 11M06

  • In this paper, we derive the asymptotic formulas $ B^*_{r, s, t}(n) $ such that

    $ \mathop{\lim} \limits_{n \rightarrow \infty} \left\{ \left( \sum\limits^{\infty}_{k = n} \frac{1}{k^r(k+t)^s} \right)^{-1} - B^*_{r,s,t}(n) \right\} = 0, $

    where $ Re(r+s) > 1 $ and $ t \in \mathbb{C} $. It is evident that the asymptotic formulas for the inverses of the tails of both the Riemann zeta function and the Hurwitz zeta function on the half-plane $ Re(s) > 1 $ are its corollaries. Subsequently we provide the asymptotic formulas for the Riemann zeta function and the Hurwitz zeta function on the half-plane $ Re(s) < 0 $. Finally, we study the asymptotic formulas of the inverse of the tails of the Dirichlet L-function for $ Re(s) > 1 $ and $ Re(s) < 0 $.

    Citation: Zhenjiang Pan, Zhengang Wu. The inverse of tails of Riemann zeta function, Hurwitz zeta function and Dirichlet L-function[J]. AIMS Mathematics, 2024, 9(6): 16564-16585. doi: 10.3934/math.2024803

    Related Papers:

  • In this paper, we derive the asymptotic formulas $ B^*_{r, s, t}(n) $ such that

    $ \mathop{\lim} \limits_{n \rightarrow \infty} \left\{ \left( \sum\limits^{\infty}_{k = n} \frac{1}{k^r(k+t)^s} \right)^{-1} - B^*_{r,s,t}(n) \right\} = 0, $

    where $ Re(r+s) > 1 $ and $ t \in \mathbb{C} $. It is evident that the asymptotic formulas for the inverses of the tails of both the Riemann zeta function and the Hurwitz zeta function on the half-plane $ Re(s) > 1 $ are its corollaries. Subsequently we provide the asymptotic formulas for the Riemann zeta function and the Hurwitz zeta function on the half-plane $ Re(s) < 0 $. Finally, we study the asymptotic formulas of the inverse of the tails of the Dirichlet L-function for $ Re(s) > 1 $ and $ Re(s) < 0 $.



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    [1] L. Xin, Some identities related to Riemann zeta-function, J. Inequal. Appl., 2016 (2016), 32. https://doi.org/10.1186/s13660-016-0980-9 doi: 10.1186/s13660-016-0980-9
    [2] D. Kim, K. Song, The inverses of tails of the Riemann zeta function, J. Inequal. Appl., 2018 (2018), 157. https://doi.org/10.1186/s13660-018-1743-6 doi: 10.1186/s13660-018-1743-6
    [3] H. Lee, J. Park, Asymptotic behavior of the inverse of tails of Hurwitz zeta function, J. Korean Math. Soc., 57 (2020), 1535–1549. https://doi.org/10.4134/JKMS.j190789 doi: 10.4134/JKMS.j190789
    [4] H. Ohtsuka, S Nakamura, On the sum of reciprocal Fibonacci number, Fibonacci Quart., 46/47 (2008), 153–159.
    [5] W. Hwang, K. Song, A reciprocal sum related to the Riemann zeta function at $s$ = 6, arXiv, 2017. https://doi.org/10.48550/arXiv.1709.07994
    [6] H. Xu, Some computational formulas related the Riemann zeta-function tails, J. Inequal. Appl., 2016 (2016), 132. https://doi.org/10.1186/s13660-016-1068-2 doi: 10.1186/s13660-016-1068-2
    [7] G. Choi, Y. Choo, On the reciprocal sums of products of Fibonacci and Lucas number, Filomat, 8 (2018), 2911–2920. https://doi.org/10.2298/FIL1808911C doi: 10.2298/FIL1808911C
    [8] T. Komatsu, On the nearest integer of the sum of reciprocal Fibonacci numbers, Aport. Mate. Inv., 20 (2011), 171–184.
    [9] H. Lee, J. Park, Asymptotic behavior of reciprocal sum of two products of Fibonacci numbers, J. Inequal. Appl., 2020 (2020), 91. https://doi.org/10.1186/s13660-020-02359-z doi: 10.1186/s13660-020-02359-z
    [10] H. Lee, J. Park, The limit of reciprocal sum on some subsequential Fibonacci number, AIMS Math., 6 (2021), 12379–12394. https://doi.org/10.3934/math.2021716 doi: 10.3934/math.2021716
    [11] D. Marques, P. Trojovsky, The proof of a formula concerning the asymptotic behavior of the reciprocal sum of the square of multiple-angle Fibonacci numbers, J. Inequal. Appl., 2022 (2022), 21. https://doi.org/10.1186/s13660-022-02755-7 doi: 10.1186/s13660-022-02755-7
    [12] Z. Xu, W. Wang, The infinite sum of the cubes of reciprocal Pell numbers, Adv. Differ. Equations, 2013 (2013), 184. https://doi.org/10.1186/1687-1847-2013-184 doi: 10.1186/1687-1847-2013-184
    [13] I. Tanackov, Ž. Stević, Calculation of the value of the critical line using multiple zeta functions, AIMS. Math., 8 (2023), 13556–13571. https://doi.org/10.3934/math.2023688 doi: 10.3934/math.2023688
    [14] T. M. Apostol, Introduction to analytic number theory, Springer, 1976. https://doi.org/10.1007/978-1-4757-5579-4
    [15] E. M. Stein, R. Shakarchi, Complex analysis, Princeton University Press, 2003.
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