On The Equation Of Several Functions Of Number Theory And Related Issues

It is well known that various times mathematicians devoted much attention to the properties of some special sequences and the arithmetic function, and obtained many significant research results on Number Theory. The new question appearance is quicker than the unsolved problem solution. Famous American nationality Romania theory of numbers theorist Florentin Smarandache in the book named "Only problems, Not solutions" introduced many interesting questions and conjectures. Many researchers have studied these sequences and functions, and obtained many value research results.I develop an interest in the function of F. Smarandache. In this dissertation we mainly study the average value of some arithmetical function and discuss solutions of several special equations concerning Pseudo Smarandache function. The major achievements are as listed below.1. Study the average value of the function LS(x) and achieve an accurate asymptoticformulate for it.2. We use the elementary methods to discuss the solvability of a function equation concerning the pseudo Smarandache function and give its all positive integer solutions.3. Analyze the properties of the pseudo Smarandache function Z(n) and the Smarandachereciprocal function S_c(n), prove that S_c(n) + Z(n) = 2n, S_c(n) = Z~*(n) + n . They haveinfinite positive integer solutions.4. We use the elementary methods to study perfect square number in the back concatenated square sequence.5. We study the convergence of this infinity series through the elementary means, and give some interesting results.6. Introduce a new number theory function and study its properties.