Mean Estimate On The Function Of Number Theory And Smarandache Problems

The mean value problems of arithmetical functions play an important role in the study of analytic number theory, and they relate to many famous number theoretic problems. Therefore, any nontrivial progress in this field will contribute to the development of analytic number theory. American-Romanian number theorist Florentin Smarandache introduced hundreds of interesting sequences and arithmetical functions, and presented many problems and conjectures in his life. In 1991, he published a book named 'Only problems, Not solutions!'. He presented 105 unsolved arithmetical problems and conjectures about these functions and sequences in it. Many researchers studied these sequences and functions from this book, and obtained important results.In this dissertation, we study the mean value problems of some important arithmetical functions and some aspect about Smarandache unsolved problems, we use Perron formula to study the sequence e_p(n) and give some asymptotic formulas while e_p(n) denotes the largest exponent of power p which divides n;we study the mean value of two new arithmetical functions D(n) and /(n);we obtain some idea about Smarandache unsolved problems:1. In this paper, we introduce two new arithmetical functions D(n) and I(n) and give two interesting asymptotic formula.2. Let p be a prime, e_p(n) denote the largest exponent of power p which divides n. In this paper, we study the properties of this sequence e_p(n) and give two interesting asymptotic formulas for3. We studied problem 9 and problem 50 of the book 'Only problems, notsolutions!'.