On Some Asymptotic Formulas In Number Theory

Let M be a given subset of nonnegative integers, SM={n|n=p1α1 p2α2…pkαkis the prime factorization of n andαi∈M for i=1,2,…,k}.For arithmetical function f(n),it is of some interest to study the mean value of f(n) while n∈SM.The value ofω(n) is defined to be the number of distinct primes dividing n. Let T(n) be the divisor function,σ(n),φ(n) denote the sum of the divisors function and the Euler's function.In this paper,by considering the case of M={0,1,b1,b2,…}(subset of nonnegative integers containing 0,1),and f (n) as unit function,2ω(n),T(n),σ(n),φ(n)and respectively,we get the asymptotic formulas of...