The Nature Of The Functions Related To The Smarandache Function

Florentin Smarandache, an American Mathematician of Romanian descent , has generated a vast variety of mathematical problems. Some problems have been solved by some number theorists from home and abroad. While other unsolved but interesting arithmetical problems, including special sequences,arithmetical functions and so on, many researchers have studied and explored them deeply, and obtained a lot of very meaningful results. It plays an irreplaceable and important role in the field of Number Theory.This dissertation is based on the study and research of the Smarandache functions. Some methods mentioned in the elementary number theory and analytic number theory are used to consider the problems which are related with the Smarandache functions, then a conjecture and an asymptotic formula are given, a special equation is solved and Smarandache-Riemann sequences are generated. Specifically, we can elaborate it from the following aspects:1. For any positive integer n, the famous F.Smarandache LCM function SL(n) is defined as the smallest positive integer k such that n | [1, 2,…, k], where [1, 2,…, k] denotes the least common multiple of all positive integers from 1 to k. Through the study and research of the F.Smarandache SL(n) function , a new arithmetical function SL~*(n), which is related with the Smarandache function SL(n), is given to study the properties of SL~*(n), a special conjecture involving function SL~*(n) is obtained, and the true conjecture is proved. Secondly , the elementary and analytic methods are used to study the mean value distribution property of (P(n) - p(n))SL{n) and an interesting asymptotic formula is given for it.2. For any positive integer n, letφ(n) and S(n) be the Euler function and the Smarandache function respectively. The elementary methods of Analytic Number Theory,Mathimatica software and C language Programming are used to study the solvability of the equation(?)meanwhile, we prove that this equation has only two positive integer solutions n = 1,10.3. The methods of the elementary number theory are used to study the property of the Smarandache-Riemannζsequence, which is generated further, a general result is obtained and a concrete proof is given.