The sum of divisors of a quadratic form

Abstract

We study the sum 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 $$S_3(X)=C_1{X}^3\log X+C_2{X}^3+O(X^2{\log }^{7}X),$$ where $C_1,C_2$ are two constants. This improves upon the error term $O_{\epsilon }(X^{8/3+\epsilon })$ obtained by Guo and Zhai.

arXiv commentAccepted by Acta Arithmetica