On The Nature Of Some Number Theoretic Function

In mathematics research, Number theory should be one of the most ancient branches, which has special status both in the modern pure mathematics and the technological research. After entering the new century,Number theory is widely applied in information system, computer science and technology, cryptology, and so on.In the contemporary research, many scholars studied some old questions involving the mean value on some arithmetic functions, and obtained some good results. They also showed their great interest on other new questions posed by professor F. Smarandache of Romania, R.K. Guy of Canada. K. Kashihara of Japan and so on.Their research has aroused many theory of numbers amateur's interest.During my study of number theory, we apply the methods of Elementary number theory and Analytic number theory to discuss several unsolved questions proposed by them. Specifically speaking, the main achievement of this article includes the following several aspects:1. Value estimate on one kind of double Tornheim L-series is discussed.In this paper, to the random Dirichlet source characteristicχ,ψ, we have given double L-and expresses L(k, l, d;χ,ψ) simple formula. Compare with Terhune double L-and the formula, when ,we obtained seal form value of L(k, l, d;χ,ψ).2. Defined two new arithmetic functions fk(n) and Its dual function fk(n) is fk(l)= fk(l)= 1k;With n> 1 and n= p₁^α1P₂^α2…p_r^αr is stan-dard decomposition type of n,defined fk(n) = max(α₁^k,α₂^k,…,α_r^k) and fk(n)= min(α₁^k,α₂^k,…,α_r^k).The two new arithmetic functions is studied by the methods of Elementary number theory and Analytic number theory, and several asymptotic formulae are obtained.