On the Volterra integral equation for the remainder term in the asymptotic formula on the associated Euler totient function

Abstract

J.Kaczorowski and K.Wiertelak considered the integral equation for remainder terms in the asymptotic formula for the Euler totient function and for the twisted Euler ϕ-function. In 2013, J.Kaczorowski defined the associated Euler totient function which extends the above two functions and proved an asymptotic formula for it. In the present paper, first, we consider the Volterra integral equation for the remainder term in the asymptotic formula for the associated Euler totient function. Secondly, we solve the Volterra integral equation and we split the error term in the asymptotic formula for the associated Euler totient function into two summands called arithmetic and analytic part respectively.