Abstract

Abstract. We nd an asymptotic formula of the sum of multi-plicative functions with Dirichlet inversion formula. 1. IntroductionIn studying prime number theorem, several functions, such as Cheby-shev function, Mangoldt function and divisor function and etc. are for-mulated as tools for the investigation of the distribution of primes. Es-pecially, the properties of Mangoldt function and zeta function give a lotof informations of prime numbers. In investigating the distribution ofprimes, Dirichlet inversion formula has been used to get an asymptoticformula for the sum of Mangoldt function. Thus it is natural to studythe sum of a certain mutiplicative function with the inversion formula.For the other multiplicative functions which we will mention in thelemmas, the general zeta functions can be de ned and they utilized inmany other problems. However, the similar results can be obtained withexactly the same manner as in the method we do in this paper, exceptdivisor problems.2. Multiplicative FunctionsA function ˙(n) de ned on integers is called mutiplicative if f(mn) =f(n)f(n) when mand nare relative prime. Since there are many appli-cations of the generalized zeta functions, it is worthwhile to study the

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