The mean square of the divisor function

Chaohua Jia ¹   ; Ayyadurai Sankaranarayanan ²

¹ Institute of Mathematics Academia Sinica Beijing 100190, P.R. China and Hua Loo-Keng Key Laboratory of Mathematics Chinese Academy of Sciences Beijing 100190, P.R. China
² School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road, Mumbai 400005, India

Acta Arithmetica, Tome 164 (2014) no. 2, pp. 181-208

Résumé

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 $$ E(x)=O(x^{1/ 2}(\log x)^5). $$

Détail

DOI : 10.4064/aa164-2-7

Keywords: divisor function ramanujan stated without proof sum leq log where cubic polynomial varepsilon varepsilon being sufficiently small positive constant stated assuming riemann hypothesis varepsilon wilson proved above result unconditionally direct application would produce log log log ramachandra sankaranarayanan proved above result without assumption paper prove log