Asymptotic Relations Between Interpolation Differences and Zeta Functions

Asymptotic relations between zeta functions (such as, $\zeta (s),\beta (s)$, and other Dirichlet $L$-functions) and interpolation differences of functions like $\mathrm{\mid }y{\mathrm{\mid }}^s$ and their interpolating entire functions of exponential type $1$ are discussed. New criteria for zeros of the zeta functions in the critical strip in terms of integrability of the interpolation differences are obtained as well.