Contains The Mean Of The Gaussian Function Equation And Its Sequence

The study on the properties of all kinds of arithmetical sequences plays a vital important role in the number theory all the time, it is also related to many famous number theoretic problems. American-Romanian number theorist Florentin Smarandache presented a lot of problems and conjectures which are related to Smarandache functions. He published a book named "Only Prob-lems, Not Solutions!" in American and presented 105 unsolved problems and related conjectures about number theory functions and sequences in this book. At the sametime, Kenichiro Kashibara doctor and Charles Ashbacher doctor in their books named "Comments and Topics on Smarandache Notions and Prob-lems" and "Collection of Problems on Smarandache Notions" both proposed a lot of number theory problems about Smarandache which have not solved yet, and many of them have good value. As this questions were presented, research interesting of many lover of number theory were triggered, and obtained some important valued results on theory.Based on the interest of above questions. In this dissertation, an equation of Gauss round function, some new Smarandache function and Prime-Digital Sub-sequence are studied by using elementary and analytic methods. All real number solutions of some related equations, some related identities and asymptotic for-mulas of sequence are obtained. The main achievements of this dissertation are as follows:1. Using elementary methods and the properties of the Gauss function [x], the solvability of x[y]-[x]y=|x-y|is discussed and its all real number solutions are obtained.2. The properties of the Smarandache Prime-Digital Subsequence andπ(n) 3. A new Smarandache function D(n) is defined and the properties of In(D(n)) and In(D(n))/n are studied by using the elementary and analytic meth-ods, and sharper asymptotic formulas are established.