An analytic approach to the remainder terms in the asymptotic formulas -- the Volterra integral equation, the Whittaker function

In the present paper, firstly, we consider the Volterra integral equation of second type for a remainder term in an asymptotic formula of an arithmetic function which satisfies some special conditions and obtained a solution of the equation. The method using there is applied to the remainder term in an asymptotic formula of the associated Euler totient function. Secondly, we consider a function defined a series over all non-trivial zeros of the generalized $L$-functions. We proved some analytic properties which satisfy some conditions. In particular, we use the Whittaker function which is kind of the confluent hypergeometric function there.