The Mean Square of Divisor Function

Let $d(n)$ be the divisor function. In 1916, S. Ramanujan stated but without proof that $\sum_{n\leq x}d^2(n)=xP(\log x)+E(x),$ where $P(y)$ is a cubic polynomial in $y$ and $E(x)=O(x^{{3\over 5}+ε}),$ where $ε$ is a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis(RH), $E(x)=O(x^{{1\over 2}+ε}).$ In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce $E(x)=O(x^{1\over 2}(\log x)^5\log\log x).$ In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we shall prove $E(x)=O(x^{1\over 2}(\log x)^5).$

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