Research On Square Free Numbers And Smarandache Functions

This dissertation focuses on some problems of elementary number theory. In thisdissertation, we will study the distribution of square-free numbers, the Smarandache func-tions and the sampling of sequence.In chapter one, we give a survey to the recent development of the distribution ofnumbers and the Smarandache functions. Meanwhile, some basic knowledge of numbertheory is introduced.In chapter two, we research the distribution of square-free numbers which only con-tain n-prime factors. Some results of the distribution of square-free numbers which onlycontainn-primefactorsandthedistributionofnumberswhichonlycontainn-primefactorsare obtained.Inchapterthree,someproblemsaboutSmarandachefunctionandPseudo-Smarandachefunction, which proposed by Zhang Wenpeng and Majumadar, are studied. Particularly,we give affirmative answers to four problems proposed by Majumadar.OurworkonsomeproblemsaboutSmarandachesquarecomplementfunction, whichproposed by Russo, are contained in chapter four. On one hand, we give affirmative an-swerstotheseproblems. Ontheotherhand, weresearchthemeanvalueofsomefunctionsinvolved the Smarandache k-th power complement function and Smarandache k-th powerresidue function. Moreover, we strength a result of Liu Miaohua.In chapter five, a new sampling, named the polynomial sampling of sequence, isdefined. We get the relationship between the polynomial and the period and linear com-plexity of the new sequence.