The Mean Value Of Multiplicative Functions In Short Intervals

Let ?(n) be the divisor function. The mean value of this function plays a very important role in number theory. Suppose that k?2, r ? 2 is an integer. Applying the result of Cui, Lii & Wu [1], we get the formula of three general mean value estimate about 1/(?(nk),1/((?(n))k) and 1/(r?(nk) in short intervals. Thus we improve previous results.This paper consists of three chapters. In the first chapter, we introduce the back-ground of some knowledge we relate to, involving some definitions, notations and the main results of this thesis. In the second chapter, we prove some lemmas, mainly the theory of multiplicative functions. The last chapter is the proof of our main theorems so that we conclude general mean value estimate of 1/(?(nk),1/((?(nn)k) and 1/r?(nk)·). We prove the following estimates.(?)and(?) hold true for (x7/12+??y?x,where An(k), Bn(k), Cn(k) are some positive constants that depend only on k and n. For k=1, 2, r=2, we improve the result of Sedunova [2],with shorter intervals.