On the error term of an asymptotic formula of Ramanujan

The mean square of the divisor function

“…Conjecture (Ramachandra-Sankaranarayanan). Assuming the RH, we have E(x) = O(x For the average situation, in 2005, H. Maier and A. Sankaranarayanan [7] proved, 1 X 2X X E 2 (x)dx ≪ X exp(−c(log X) 3 5 (log log X) − 1 5 ), (1.10) where c is a positive constant.…”