On 15/07/2020 16:59, Neil Sloane wrote:
A018805(n) is no. of pairs (x,y) in [1..n]X[1..n] with gcd(x,y)=1. It is ~ 6*n^2/Pi^2 + C*n*log n, I think. What is C, and how can one find it using Mathematica or some other program? I ask because there are several similar sums I am interested in. Is there a reference?
According to an answer to https://math.stackexchange.com/questions/64498/probability-that-two-random-n... Mertens proved in 1874 that the error is O(n log n); Walfisz proved in 1963 that it's O(n (log n)^2/3 (log log n)^1/3); Chowla and Pillai proved in 1930 that it isn't o(n log log log n) and that the sum of the error terms for n=1, ..., n=N is asymptotically 3/pi^2 N^2; that Montgomery proved in 1987 that the error is infinitely often at least of order n sqrt(log log n). More details from the author of that answer at https://enaslund.wordpress.com/2012/01/15/the-sum-of-the-totient-function-an... So it looks as if C=0 and the actual magnitude of the error term isn't fully known. -- g