The mean square of the divisor function

Chaohua Jia, and Ayyadurai Sankaranarayanan. "The mean square of the divisor function." Acta Arithmetica 164.2 (2014): 181-208. <http://eudml.org/doc/278861>.

@article{ChaohuaJia2014,
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 $E(x) = O(x^\{3/5 + ε\})$, with ε being a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis (RH), $E(x)=O(x^\{1/2 + ε\})$. In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce $E(x) = O(x^\{1/2\}(log x)⁵loglog x)$. In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we prove $E(x) = O(x^\{1/2\}(log x)⁵)$.},
author = {Chaohua Jia, Ayyadurai Sankaranarayanan},
journal = {Acta Arithmetica},
keywords = {divisor function; Riemann zeta function; mean value},
language = {eng},
number = {2},
pages = {181-208},
title = {The mean square of the divisor function},
url = {http://eudml.org/doc/278861},
volume = {164},
year = {2014},
}

TY - JOUR
AU - Chaohua Jia
AU - Ayyadurai Sankaranarayanan
TI - The mean square of the divisor function
JO - Acta Arithmetica
PY - 2014
VL - 164
IS - 2
SP - 181
EP - 208
AB - 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 $E(x) = O(x^{3/5 + ε})$, with ε being a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis (RH), $E(x)=O(x^{1/2 + ε})$. In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce $E(x) = O(x^{1/2}(log x)⁵loglog x)$. In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we prove $E(x) = O(x^{1/2}(log x)⁵)$.
LA - eng
KW - divisor function; Riemann zeta function; mean value
UR - http://eudml.org/doc/278861
ER -