. 2010, 28(17): 50-53.
For any positive integern, the Smarandache power function SP(n) is defined as the smallest positive integermsuch that mm is
divisible by n. The main purpose of this paper is to study the solvability of equations SP(n)=?准(n), k=1, 2, 3 (for the Euler function) and all the positive integer solutions and other related issues, based on the nature of the series{SP(n)}, 1st mean, asymptotic formula, the
convergence of infinite series SP(n) and its related identity. The analytic methods are used to get the distribution properties of the k-th
Powers of SP(n). For any real number x≥3, given the real numbers k, l (k>0, l≥0), and all the prime numbers p, any positive number ε
and the Riemann Zeta-function, we give and prove the corresponding asymptotic formula. For any real number x≥3 and a given real number k′>0, we also give and prove the corresponding asymptotic formula. For any given real numbers x≥3 and real numbers l≥0, the corresponding asymptotic formula is also given and proven together. Thus, the asymptotic formula of ■nl(SP(n))k and ■■(k>0, l≥0) is given, when l=0, k=1/k′, k=1,2,3 and ζ(2)=π2/6, ζ(4)=π4/90, and it could be found that the theorem is a further extension of the related results.
