Rings Of Number-theoretic Functions On Multiplicative Subsemigroups Of Positive Integers

Classical number-theoretic functions are functions from the positive integers to the complex numbers. Such functions play a key role in Number Theory. Many number-theoretic problems can be translated into the problems of number-theoretic functions. Usu-ally such problems lead to the estimations of orders, bounds or average orders of some number-theoretic functions.The general theory of number-theoretic functions defined on multiplicative subsemi-groups of positive integers is not complete. Only some special functions have been studied. The purpose of this thesis is to generalize the theory of classical number-theoretic func-tions to that of generalized ones. After that a criterion for judging the convergence modes of a special kind of series is given, and two asymptotic estimations on the distribution of elements of multiplicative subsemigroups of N+in N+is provided.In the first chapter the historical background and research status of the theory of number-theoretic functions are introduced and some examples of relating classical number-theoretic problems to such functions are given. Next the divisibility theory of monoids is reviewed and is applied to the study on the structures of multiplicative subsemigroups of N+.In the second chapter, a theory of generalized number-theoretic functions defined on multiplicative subsemigroups of N+is developed. Then an important class of functions are studied. Some classical functions are generalized to the corresponding functions in gener-alized number-theoretic functions’ring. Let ε, ε’be arbitrary, ε< ε’, and S={s1, s2,...} satisfies special conditions. Denote<S> as multiplicative subsemigroups generated by S. The following is obtain:if Skte, then the series ∑ns>1/nε’ converges, and if Sk k1/ε then the series ∑n<S>1/nε diverges. Another result:if Sk k1/εg(k), in which g(k)= o(kα) for any α> 0, and g(k)1, and C<s>(X):=∑n<s>:n<x1.Then for any δ>0, x≥s1, the following holds: Xε-ε＜＜ C<s>(x) ＜＜ xε+δ.In the third chapter the properties of generalized Mobius function are studied deeply and a formula is given. Using this formula an asymptotic estimation on the distribution of elements of finitely generated multiplicative subsemigroups of N+in N+is deduced:Let finite set S={p1,p2, ...Pt｝ N+, and suppose the elements of S are all primes in N+, then C<s>(x)= c (log x)t+O ((log xt-1), in which the constant c depends only on S.