The Mean Value Of Special Multiplicative Functions

Arithmetic functions play an important role in the study of number theory problem,and they are related to many famous number theory problems.Although a single value of many important arithmetic function is very irregular,but their average results reflect good regularity.Therefore,nature of the arithmetic function in number theory is often conducted in average sense.Let f be a nonnegative multiplicative function which satisfies f(n):＜＜ nε and f(pl)《Al,where n,l ∈ N,A +R+,p is a prime and ε>0 arbitrarily small.In this paper,by the prime number theorem and smooth number theory,the mean value of f(n2)was established,which generalized the work of Shiu[3].Secondly,we consider the result in arithmetic progressions in short intervals.Finally,an estimation on the Fourier coefficient of cusp forms was given by the formula.This article mainly focuses on the average estimate on a class of special function,the article will be divided into five chapters.In the first chapter,we briefly introduce the development history of number theory,some definitions and notations,and the main result of the paper.In the second chapter,we introduce several lemmas which will be used in this paper.In the third chapter,the main theorem is proved.And we give the proof of the result in arithmetic progressions.