The Mean Values Of Pillai’s Arithmetical Function

Pillai’s arithmetical function(The gcd-sum function) is defined bywhere gcd(a, b) denotes the greatest commom divisor of a and b.In1933, Pillai proved the following conclusionwhere is Euler’s function, σ(n) denotes the sum of divisors of n.The convolution method applied for P (n) leads to the asymptotic formulaIn1985, Chidambaraswamy and Sitaramachandrarao proved that, given anarbitrary>0,where A1, A2are computable constant and0<θ <1/2is some exponent containedin Dirichlet divisor problemL.T′oth first proved the mean value of the reciprocal of P (n) Shiqin Chen and Wenguang Zhai further improved the results and obtainedasymptotic formulawhere N is a constant§Kjare computable constants.The arithmetic mean of gcd(1, n),..., gcd(n, n) is given byL.T′oth proved the following conclusion, for any>0,where C1, C2, C3, C4are constants.Divisor problem for the function d4(n)=1, is thatfor any>0, where K1=16, K2, K3, K4ate constants. Assume that α1with a positive constant c.If the Riemann hypothesis is true, then for any real x sufciently large, theerror term isIf α4is near38and RH is true, then the exponentIn this paper, we study mean value of Pillai’s arithmetical function, there aretwo main results as following:Theorem1If the Riemann hypothesis is true, for any>0where P3(log x) is a polynomial of degree3in log x.Theorem2Suppose N>1is a fixed integer, then where kjare computable constants.In this paper, we study function A(n) sum over k-full integer, If prime p isdivisor of n, then pk|n, the standard composition formula isFor each positive integer n, if n is k-full integer, define δk(n)=1, otherwisewe call δk(n) is the character function of k-full integer.DefineWe have the following resultsTheorem3For any>0,Pk(log x) is a polynomial of degree k in log x.Theorem4For any>0,S3(x)=x13(P3(log x))+x14(P4(log x))+x15(P5(log x))+O(xσ3+),wherePk(log x) is a polynomial of degree k in log x.We also study the short interval case of A(n) and proved the followingTheorem5If x14311where is the Riemann Zeta function.