@article{Pan2024, 
author = {Zhenjiang Pan and Zhengang Wu},
title = {The inverse of tails of Riemann zeta function, Hurwitz zeta function and Dirichlet L-function},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {6},
pages = {16564-16585},
keywords = {Riemann zeta function, Hurwitz zeta function, reciprocal sums, asymptotic formulas, Dirichlet L-function},
url = {https://www.sciopen.com/article/10.3934/math.2024803},
doi = {10.3934/math.2024803},
abstract = {In this paper, we derive the asymptotic formulas        B          r      ,      s      ,      t        ∗    (  n  ) such that               lim              n      →      ∞        ⁡      {                            (                                    ∑                              k                =                n                                            ∞                                                    1                                                k                  r                                (                k                +                t                                  )                  s                                                              )                          −          1                    −              B                  r          ,          s          ,          t                ∗            (      n      )        }    =  0  ,where    R  e  (  r  +  s  )  >  1 and    t  ∈      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    R  e  (  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    R  e  (  s  )  <  0. Finally, we study the asymptotic formulas of the inverse of the tails of the Dirichlet L-function for    R  e  (  s  )  >  1 and    R  e  (  s  )  <  0.}
}