About Addition Means Function And Continuous Excess

Let A be an infinite set of positive integers. For any positive integer n, let r(A, n) be the number of solutions of the equation n = a+b with a,b?A,a?b.Let |A(x)| be the number of integers in A which are less than or equal to x. In 1998, Nicolas, Ruzsa and (?) proved that there exists an infinite set A of positive integers such that r(A, n)?1 for all sufficiently large integers n and (?) |A(x)|(logx)-2 ? (log 2)-2. They also proved that, if A is an infiniteset of positive integers such that r(A, n)? 1 for all sufficiently large integers n, then (?)|A(x)|(loglogx/logx)3/2 ? 1/20. In 2004, Balasubramanian and Prakash showed that there exists an absolute constant c > 0 with the following property: for any infinite set A of positive integers with r(A,n) ? 1 for all suffi-ciently large integers n, we have |A(x)|?c (logx/toglogx)2 for all sufficiently large x. In this paper, we prove that, if r(A,n)?A 1 for all sufficiently large integers n, then |A(x)|>1/2(lonx/loglogx)2 for all sufficiently large A.Let ?(n) be the sum of all positive divisors of n, E (x) be the largest number of consecutive numbers n not exceeding x with ?(n) ? 2n. In 1935, P. Erdos proved that c1 log log logx ?E(x)?c2logloglogx, where c1 and c2 are two positive constants. In 2011, Pollack proved that E(x)/logloglogx tends to a limit as x ? +? by means of the theory of real variable function. In this thesis,we present a different approach to this old problem by using the knowledge of number theory.