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

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