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 $$S_3(X)=C_1X^3\log X+ C_2X^3+O(X^2\log^7 X),$$ 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.

The sum of divisors of a quadratic form

Let $\tau_3(n)$ be the triple divisor function which is the number of solutions of the equation $d_1d_2d_3=n$ in natural numbers. It is shown that $$ \sum_{1\leq n_1,n_2,n_3\leq \sqrt{x}}\tau_3(n_1^2+n_2^2+n_3^2)=c_1x^{\frac{3}{2}}(\log x)^2+ c_2x^{\frac{3}{2}}\log x +c_3x^{\frac{3}{2}} +O_{\varepsilon}(x^{\frac{11}{8}+\varepsilon}) $$ for some constants $c_1$, $c_2$ and $c_3$.

Sums of the triple divisor function over values of a ternary quadratic form

Let f(x1, . . . , xs) (s ≥ 3) be a regular quadratic form with integral variables. We study the number of divisors of f(x1, . . . , xs) on average. We establish an asymptotic formula of the sum of these divisors.

A Divisor Problem Attached to Regular Quadratic Forms

For an integer , let denote the ordinary divisor function. This paper studies the asymptotic behavior of the sum . It is proved in the paper that, as , where and are certain constants and is any fixed positive real number. The result corrects a false formula given in a paper of Gafurov concerning the same problem, and improves the error claimed by Gafurov.

On the Number of Divisors of the Quadratic Form m2 + n2

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On the Number of Divisors of the Quadratic Form m2 + n2