On the solution of the Volterra integral equation of second type for the error term in an asymptotic formula for arithmetical functions

Abstract

In 2010, J.Kaczorowski and K.Wiertelak considered the Volterra integral equation of second type for the remainder term in the asymptotic formula for the Euler totient function. The author found that the consideration made by them holds for other remainder terms in the asymptotic formula having certain common properties. In this paper, we first consider the pair of complex-valued arithmetical functions $(a(n),b(n))$ satisfying $b(n) \hspace{-0.1cm} = \sum_{d|n} a(d)n/d$. We prove that the solution of the Volterra integral equation of second type for the error term in the asymptotic formula for $b(n)$ can be obtained when $a(n)$ satisfies some special condition.