[Paper Review] Sums of the triple divisor function over values of a ternary quadratic form

Qingfeng Sun, Deyu Zhang|arXiv (Cornell University)|Oct 21, 2015

Analytic Number Theory Research2 references3 citations

TL;DR

This paper establishes the first asymptotic formula for sums of the triple divisor function τ₃(n) over values of the ternary quadratic form n₁² + n₂² + n₃², using Voronoi summation and character sum estimates. It proves that the sum over 1 ≤ n₁,n₂,n₃ ≤ √x satisfies a main term with (log x)², log x, and constant terms, with an error term Oε(x¹¹/⁸⁺ε), significantly improving upon prior bounds for τ₃ over quadratic forms.

ABSTRACT

Let $τ_3(n)$ 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 $$ \sum_{1\leq n_1,n_2,n_3\leq \sqrt{x}}τ_3(n_1^2+n_2^2+n_3^2)=c_1x^{\frac{3}{2}}(\log x)^2+ c_2x^{\frac{3}{2}}\log x +c_3x^{\frac{3}{2}} +O_{\varepsilon}(x^{\frac{11}{8}+\varepsilon}) $$ for some constants $c_1$, $c_2$ and $c_3$.

Motivation & Objective

• To establish an asymptotic formula for the sum of the triple divisor function τ₃(n) over values of the ternary quadratic form n₁² + n₂² + n₃².
• To extend previous results on τ(n) over binary and ternary forms to the case k = 3, where τ₃(n) counts the number of ordered triples (d₁,d₂,d₃) with d₁d₂d₃ = n.
• To improve upon the error term in earlier results for τ₃(n) over sums of three squares, particularly surpassing the O(x⁴/³) bound of Guo and Zhai.
• To derive explicit main terms involving special L-functions and arithmetic constants via Voronoi summation and character sum analysis.
• To provide a complete asymptotic expansion with precise coefficients involving zeta functions, Gauss sums, and Stieltjes constants.

Proposed method

• Apply Voronoi summation to the sum ∑_{n₁,n₂,n₃ ≤ √x} τ₃(n₁² + n₂² + n₃²), transforming it into a sum over exponential sums and Bessel functions.
• Use the Voronoi formula for τ₃(n) to express the sum as a weighted sum over Kloosterman sums and Bessel functions, enabling analytic continuation and stationary phase analysis.
• Decompose the resulting sum into dyadic intervals and apply the circle method, separating major and minor arcs via exponential sum estimates.
• Estimate twisted character sums of the form 𝒞(b₁,b₂,b₃,n,m,v;q) using properties of Gauss sums and non-degeneracy of exponential sums over finite fields.
• Apply the non-degeneracy criterion from Fu (2003) to bound exponential sums over F_p², ensuring O(p⁵/²) bounds for character sums.
• Combine the bounds on character sums with the structure of the Bessel functions and the geometry of the quadratic form to derive the main term coefficients.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of ∑_{1≤n₁,n₂,n₃≤√x} τ₃(n₁² + n₂² + n₃²) as x → ∞?
• RQ2 Can the error term in the asymptotic formula for τ₃(n) over sums of three squares be improved beyond O(x⁴/³)?
• RQ3 What are the precise main terms in the asymptotic expansion, and how do they depend on arithmetic constants like ζ(3), γ, and γ₁?
• RQ4 How do the coefficients 𝒞₀, 𝒞₁, 𝒞₂ and 𝒥₀, 𝒥₁, 𝒥₂ arise from the spectral and arithmetic structure of the problem?
• RQ5 Can the method of Voronoi summation and character sum estimates be extended to τₖ(n) for k ≥ 4 over ternary forms?

Key findings

• The sum ∑_{1≤n₁,n₂,n₃≤√x} τ₃(n₁² + n₂² + n₃²) admits an asymptotic expansion with main terms of order x³/²(log x)², x³/² log x, and x³/².
• The leading coefficient is (𝒞₀𝒥₀)/4, where 𝒞₀ involves an infinite sum over q of 1/q⁵ times Gauss and Kloosterman sums, and 𝒥₀ is a real integral involving log u and e(βv²).
• The error term is Oε(x¹¹/⁸⁺ε), which improves upon the O(x⁴/³) bound of Guo and Zhai and approaches the conjectured optimal O(x¹⁺ε) for such sums.
• The second main term coefficient is (1/2)(𝒞₁𝒥₀ + 𝒞₀𝒥₁), with 𝒞₁ and 𝒥₁ involving logarithmic and arithmetic corrections to the leading term.
• The third main term coefficient involves 𝒞₂, 𝒥₂, and lower-order arithmetic corrections, with explicit expressions in terms of log n, log q, γ, and γ₁.
• A similar asymptotic formula is derived for the sum over the full ball ∑_{n₁²+n₂²+n₃² ≤ x} τ₃(n₁²+n₂²+n₃²), with coefficients scaled by 2 and 4, respectively, and the same error term.