The sum of divisors of a quadratic form
Volume 163 / 2014
Acta Arithmetica 163 (2014), 161-177
MSC: Primary 11P05; Secondary 11P32, 11P55.
DOI: 10.4064/aa163-2-6
Abstract
We study the sum $\tau$ of divisors of the quadratic form $m_1^2+m_2^2+m_3^2$. Let $$S_3(X)=\sum_{1\le m_1,m_2,m_3\le X}\tau(m_1^2+m_2^2+m_3^2).$$ We obtain the asymptotic formula $$S_3(X)=C_1X^3\log X+ C_2X^3+O(X^2\log^7 X),$$ where $C_1,C_2$ are two constants. This improves upon the error term $O_\varepsilon(X^{8/3+\varepsilon})$ obtained by Guo and Zhai (2012).
