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

In this paper, we derive the asymptotic formulas $B_{r,s,t}^{\ast }(n)$ such that

$$\underset{n\to \mathrm{\infty }}{lim}\left\{{\left(\sum _{k=n}^{\mathrm{\infty }}\frac{1}{k^r(k+t{)}^s}\right)}^{-1}-B_{r,s,t}^{\ast }(n)\right\}=0,$$

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 $Re(s)<0$.