The mean square of the divisor function
Volume 164 / 2014
Acta Arithmetica 164 (2014), 181-208
MSC: Primary 11M; Secondary 11M06.
DOI: 10.4064/aa164-2-7
Abstract
Let $d(n)$ be the divisor function. In 1916, S. Ramanujan stated 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 $$ E(x)=O(x^{{3/ 5}+\varepsilon}), $$ with $\varepsilon$ being a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis (RH), $$ E(x)=O(x^{{1/ 2}+\varepsilon}). $$
In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce $$ E(x)=O(x^{1/ 2}(\log x)^5\log\log x). $$ In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption.
In this paper, we prove $$ E(x)=O(x^{1/ 2}(\log x)^5). $$
