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
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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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
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