  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.
