Smarandache Function Mean And The Solvability Of Study

What is the number theory? Number theory is a mathematical branch that mainly focuses on rules of numbers, especially the property of integers. As same as geometry, it is the oldest and is always attracting researcher's attention. Nowadays, the number theory is one of the earliest and fastest developed branch of mathematics, and the study of the mean value and properties of some arith-metical functions and sequences is very important in number theory in China. In 1993, American-Romanian number theorist Florentin Smaradache published a book named "Only Problems, Not Solutions!". In this book, professor Smaran-dache presented 105 unsolved arithmetical problems and conjectures about spe-cial sequences and arithmetical functions. As a result, many subsequent studies were carried on about the sequences and functions, and some important number theories were uncovered.The thesis mainly focuses on the convergence and divergence properties of Smarandache functions and properties of some special functions, using ele-mentary and analytical methods to study the mean value of theκ-th Smaran-dache dual function Sk(n), and the solutions of the equation involving theκ-th Smarandache dual function Sk(n), and the general formula of the Smarandache function E(n,r) and O(n,r). Specifically, here are the achievements based on the research:1. Based on the previous studies of the Smarandache function, elementary and analytical methods are both used to study the mean value of function Sk(n), then we get some interesting results and the average estimate equation through conjecture, conclusion and tidy up.2. Using the elementary methods to study the calculating problem of the equation The definitions ofω(n) andΩ(n) are as follows:ω(n) =r, Q,(n)=α1+α2+···+αr. Which n = p1α1pp2αr denotes the factorization of n into prime powers, then we get all the solutions of this equation.3. Using the elementary methods to study the general formula of E(n, r) and O(n, r), E(n, r) are the sums of products of first n even natural numbers, taken r at a time, r< n. O(n,r) are the sums of products of first n odd natural numbers, taken r at a time, r≤n, they are also without repeatition.