The Mean Value Of The Index Of Composition Of An Integer In The Set Of Piatetski-Shapiro Prime

The paper contains two chapters. We study the mean value of the index of composition of an integer in the set of Piatetski-Shapiro prime and estimates of the e-squarefree e-divisor function in short intervals.In chapter 1, we study the mean value ofλ1(n) in the set of Piatetski-Shapiro prime, where n> 2.For each integer n> 2, letλ(n):=(logn)/(logγ(n))be the index of composition, whereγ(n):=∏p|nP.We writeλ(1)=γ(1)= 1. The index of composition of an integer measures the multiplicity of its prime factors. De Koninck and Doyon firstly studied the mean value ofλ(n). They proved the asymptotic formulas and where c=∑(log p)/(p p(p-1))≈75536. These two asymptotic formulas imply that the average order ofλ(n) is 1.De Koninck and Katai proved a series of results about the mean value ofλ(n). They proved that the asymptotic formulas and fbr y=x1/5 log3(x).When y=(?),they proved that for any fixed integer r≥1, there exist computable constants c1,…,cn,d1,…,dr,such that and Then they deduced and where cj,dj(j≥1)are computable constants.Zhai Wenguang improved the results of De Koninck and Katai,using the Selberg’s method(see Iivc,chapter 14)studied the higher moments ofλ(n),and proved the asymptotic formulas and whereLV Meimei improved the results of Zhai Wenguang,studied∑n≤xλk(n) higher moments ofλ(n),and proved the following results.Let k≥1 be fixed integer, then where c> 0 is the positive constant, and ck,j1, (j1=1,2,…, k), ck,j2, (j2 1,2,…,k), are computable constants. Let△(x)= x1/2e-clog3/5x(loglogx)-1/5, if the R.H. is true, then△(x)= O(x5/(14)+∈).E. Wirsing firstly studied the three prime theorem in sparse set, he proved that a sufficientily large odd number can be written as a sum of three primes of some set S, which satifies But we do not know much about the nature of the set S. A. Blog and J. Frielander successfully replaced S by so-called Piatetski-Shapiro prime set.Let[θ] denotes the integer part ofθ. Piatetski-Shapiro firstly considered the prime problem which is named by him, and studied the distribution of primes in sequence of the form [nc], he proved that for 1< c< (12)/(11). Since then, the range of c has been improved by many people (see [4-8]), Rivat and Sargos proved for 1< c< (2817)/(2426), which is the best result by now. In addition, Rivat proved for 1< c<7/6, This result was improved to 1< c<(20)/(17) in the paper [10]. Jia improved 1< c<(13)/(11) by screening method.For a fixedγ, letγ=1/c, we have for (2426)/(2817)<γ< 1. So the primes of the form [n1/γ] form a Piatetski-Shapiro prime set of orderγ. In chapter 1,we study the mean value ofλ-1(n)in the set of Piatetski-Shapiro prime,we can arrive at the following result:定理1 If 1<c<(181)/(154)=1.1753247…,then whereIn chapter 2,we study the estimates of e-squarefree e-divisor function in short interval.We want to find y as small as possible,such that the above sum in short interval has an asympotic formula. x>0 is a sufficiently large number, 0<y=o(x).Using Perron formula,we obtain the main term of the above sum. Moreover,by the convolution method,we estimate the error term.In this chapter, we arrive at the following result:定理2 If x1/5|2∈≤y≤x,then where c1:=∏p(1+∑α=6∞(2ω(α)-2ω(α-1))/(pα)).