The mean square of the divisor function
Chaohua Jia ; Ayyadurai Sankaranarayanan
Acta Arithmetica, Tome 166 (2014), p. 181-208 / Harvested from The Polish Digital Mathematics Library
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Let d(n) be the divisor function. In 1916, S. Ramanujan stated without proof that ∑n≤xd²(n)=xP(logx)+E(x), where P(y) is a cubic polynomial in y and E(x)=O(x3/5+ε), with ε being a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis (RH), E(x)=O(x1/2+ε). In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce E(x)=O(x1/2(logx)⁵loglogx). In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we prove E(x)=O(x1/2(logx)⁵).

Publié le : 2014-01-01
EUDML-ID : urn:eudml:doc:278861 
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Chaohua Jia; Ayyadurai Sankaranarayanan. The mean square of the divisor function. Acta Arithmetica, Tome 166 (2014) pp. 181-208. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa164-2-7/