The Mean Square of Divisor Function

Abstract

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

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