The Mean Square of Divisor Function

2013/11/16 by Jia, Chaohua, Sankaranarayanan, Ayyadurai
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1311.4041

Abstract

Let d(n) be the divisor function. In 1916, S. Ramanujan stated but 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\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)⁵loglog 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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