The mean square of the divisor function

Abstract

Let d(n) be the divisor function. In 1916, S. Ramanujan stated without proof that n≤x+d2(n) + xP(log x) + E(x), where P(y) is a cubic polynomial in y and E(x) = O(x3/5+∈), with ∈ being a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis (RH), E(x) = O(x1/2+∈), In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce E(x) = O(x1/2(log x)5log log x). In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we prove E(x) = O(x1/2(log x)5).