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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