The Average Estimate Of The Divisor Function Of Integer Matrices

Using the classical method in analytic number theory,this paper investigates the mean value of the divisor function of integer matrices on square-free numbers,and establishes an asymptotic formula,which generalize the related result.Let Mk(Z)denote the ring of k × k matrices over Z.We denote the number of different representations of matrix C in the form C=A1A2,C,A1,A2?Mk(Z)by?(k)(C).Many mathematicians studied the mean value of ?(k)(C).G.Bhowmik and H.Menzer[2]studied the distribution of values of the function t(2)(n)=??(2)(C),and proved T2(x):=?t(2)(n)=xP2(log x)+?2(x)n?x where ?2(x)<<x31/43+?,P2(u)is the second-degree polynomial in u.Up to now the best estimate of ?2(x)has been obtained by G.Bhowmik and J.Wu[4]?2(x)dx=O(x3(log x)31/3),A.Ivic[11]gave bounds for second moment of error term ?2(x)?1x ?22(x)dx=O{x2(log x)31/3),Study of the distribution of function t(k)(n)for k?3 has some difficulties.N.Fugelo and I.Velichko[10]constructed the generating Dirichlet series for t(3)(n)and obtained the asymptotic formulaUsing Perron's formula,I.N.Velichko[30]studied the distribution of function t(4)(n),and obtained estimates of functions T4(x)and Tk*(x),i.e.and where ?' indicates that the summation runs over all square-free numbers,P2(u)is a polynomial in u of degree 2.1.M.Velichko[29]established the asymptotic formula for a summatory function of the number of representations of matrices from M2(Z)in the form C=A1A2A3:and estimated the error term of this asymptotic formula where ?2,3(x)=?t3(2)(n)-xP5(log x).In this paper,we study the distribution of ?t3(2)(n)on the square-free num-n?x bers.In detail,we have the following result.Theorem 1 As x??,the asymptotic equality holds,where the sum ?' indicates that the summation runs over square-free num-bers.Theorem 2 Let ?2,3*(x)=?n?x't3(2)(n)-xP2(lop x),where P2(logx)is the polynomial in Theorem 1,we have holds.