Wang: Sums of cubes and the Ratios Conjectures

Full paper in Markdown.

---

Full paper in Markdown.

Victor Y. Wang, "Sums of cubes and the Ratios Conjectures," arXiv:2108.03398 (2021).

Overview

Victor Y. Wang, Sums of cubes and the Ratios Conjectures, arXiv:2108.03398 (2021), studies the critical six-variable cubic equation

$$F(\mathbf x)=x_1^3+\cdots+x_6^3=0$$

and its connection, through additive energy, to values of $F_0(x,y,z)=x^3+y^3+z^3$. For a cubic form $F$, the weighted count $N_{F,w}(X)$ is defined in (1.2), while the error $E_{F,w}(X)$, after subtracting the singular-series term and contributions from rational linear spaces on $F=0$, is defined in (1.3). The Hooley–Manin prediction is the asymptotic statement $E_{F,w}(X)=o(X^3)$, equation (1.5). The motivating Heath-Brown conjecture that every fixed $a\not\equiv\pm4\pmod 9$ has infinitely many signed three-cube representations is recalled in §1, citing [37], p. 623.

The earlier conditional benchmark, Theorem 1.1, due to Hooley and Heath-Brown, gives $N_F(X)\ll_\varepsilon X^{3+\varepsilon}$ for diagonal cubic forms in six variables, assuming automorphy and GRH for the Hasse–Weil functions $L(s,V_{\mathbf c})$. Wang’s first main result removes the epsilon in the equal-coefficient case: under Conjectures 1.2, 1.4, and 1.5, Theorem 1.3 proves

$$N_{x_1^3+\cdots+x_6^3}(X)\ll X^3$$

(equation (1.8)). It also proves that if $S\subseteq\mathbb Z_{\ge0}$ has positive lower density, then $F_0(S^3)$ has positive lower density. This is a conditional theorem, not an unconditional density result.

The hypotheses have distinct roles. Conjecture 1.2 (HW2) asserts automorphy and absence of zeros in $\Re s>1/2$ for the Hasse–Weil functions listed in (1.7). Conjecture 1.4 (R2′), equation (1.10), is a log-free second-moment estimate over $\mathbf c$ for the mollified reciprocal

$$\Phi^{\mathbf c,1}(s)=\{\zeta(2s)L(s+1/2,V)L(s,V_{\mathbf c})\}^{-1}$$

from (1.9). Conjecture 1.5 is a square-free sieve assertion for the discriminant polynomial $\Delta(\mathbf c)$. Conjecture 6.3 is the paper’s explicit two-ratios prediction; Proposition 6.8 shows that Conjectures 1.2 and 6.3 imply Conjecture 1.4. Proposition 6.1 supplies the local first- and second-moment calculations, notably (6.6)–(6.7), underlying the Ratios Recipe.

A stronger first-moment hypothesis gives asymptotics. Under Conjectures 1.2, 1.4, 1.5, and 1.8, Theorem 1.6 proves (1.5) for diagonal six-variable cubics and weights supported away from the coordinate hyperplanes, deduces the Hasse principle for $F=0$, and, for $F=x_1^3+\cdots+x_6^3$, proves that 100% of integers $a\not\equiv\pm4\pmod9$ belong to $F_0(\mathbb Z^3)$. Corollary 1.7 removes the support restriction for this equal-coefficient form. These conclusions concern signed cubes. Under the effective Ratios estimate Conjecture 1.10 and the effective local-constancy Conjecture 1.11, Theorem 1.9 strengthens the asymptotic to $E_{F,w}(X)\ll X^{3-\delta}$ for some $\delta>0$.

The proof begins with the delta-method identity (2.10), whose arithmetic factors are the complete sums $S_{\mathbf c}(n)$ from (2.8) and whose archimedean factors are $J_{\mathbf c,X}(n)$ from (2.9). The singular locus $\mathcal S_0=\{\Delta(\mathbf c)=0\}$ and smooth locus $\mathcal S_1$ are defined in (1.6). The imported unconditional Theorem 2.5 evaluates the $\mathcal S_0$-contribution as the singular-series term plus the linear-space terms, with error $O(X^{2.75+\varepsilon})$, equation (2.16); the new analysis therefore concentrates on $\mathcal S_1$.

On $\mathcal S_1$, §7 separates good and bad primes through (7.1)–(7.2) and factors the good-prime series into the three factors of Definition 7.1. Proposition 7.2 shows that the third, error factor is absolutely convergent already for $\Re s>1/3$. Propositions 6.13–6.14 and 7.15–7.16 convert the conjectural $L$-function statistics into estimates adapted to delta-method sums, including localization in residue classes. The exceptional residue-class construction is given in Definitions 7.7–7.8 and controlled qualitatively by Lemma 7.12 and effectively, assuming Conjecture 1.11, by Lemma 7.13.

Two further inputs address losses not controlled by GRH alone. Proposition 8.1 gives uniform, log-free bounds for derivatives of $J_{\mathbf c,X}(n)$, with decay governed by both $\|\mathbf c\|/X^{1/2}$ and $X\|\Delta(\mathbf c/Z)\mathbf c\|/n$. Lemma 9.1 proves vanishing and boundedness criteria for bad-prime sums $S_{\mathbf c}(p^\ell)$. For diagonal forms, Proposition 9.9 derives the geometric moment estimate Conjecture 9.8 from the square-free sieve Conjecture 1.5.

The endgame is explicit. Theorem 10.5 combines Hölder estimates with the delta decomposition to prove the epsilon-free absolute bound (10.14); §10.2 then derives Theorem 1.3 by the dyadic argument (10.20)–(10.23). Theorem 10.7 obtains cancellation over $\mathbf c$, yielding $\Sigma^\natural(X,\mathcal S_1)=o(X^{(6-m)/4})$, and is used in §10.3 to prove Theorem 1.6. Theorem 10.8 supplies a power saving and leads to Theorem 1.9. Thus the paper’s global conclusions are conditional, while many of its delta-method, local, geometric, and oscillatory-integral estimates are unconditional components of the conditional argument.

Relation to E940

This source bears on Problem 940.

Let

$$\mathcal P_r=\{n\ge1: p^e\parallel n\Rightarrow e\ge r\}$$

be E940’s set of positive $r$-powerful integers, and let $\mathcal A_r=\mathcal P_r\cup(\mathcal P_r+\mathcal P_r)\cup(\mathcal P_r+\mathcal P_r+\mathcal P_r)$ when $r=3$. Every positive cube is 3-powerful, so

$$\{x^3+y^3+z^3:x,y,z\ge1\}\subseteq\mathcal A_3.$$

Consequently, the second assertion of Theorem 1.3, applied to $S=\mathbb Z_{\ge1}$, conditionally gives positive lower density for a subset of $\mathcal A_3$. Under Conjectures 1.2, 1.4, and 1.5, E940’s density-zero assertion therefore fails already at $r=3$. Because these conjectures are unproved, this is only a conditional refutation and does not resolve E940.

The precise bridge is additive energy. For

$$B_X=\{x^3+y^3+z^3:1\le x,y,z\le X\},$$

Cauchy–Schwarz bounds $|B_X|$ below by the square of the number of triples divided by the number of equal-sum pairs. Such pairs satisfy

$$x_1^3+x_2^3+x_3^3=y_1^3+y_2^3+y_3^3,$$

which becomes $u_1^3+\cdots+u_6^3=0$ after replacing the three $y$-variables by their negatives. Hence the energy is bounded by $N_F(X)$ for $F=u_1^3+\cdots+u_6^3$. Theorem 1.3’s bound $N_F(X)\ll X^3$ gives $|B_X|\gg X^3$; since $B_X\subseteq[3,3X^3]$, this is the natural positive-density scale. Thus an unconditional proof of the special estimate (1.8)—or a sufficiently strong substitute controlling this energy—would furnish a direct route to refuting E940 at $r=3$.

Theorem 1.6’s 100% result is less directly usable for E940: it concerns signed representations $a=x^3+y^3+z^3$ with $x,y,z\in\mathbb Z$, whereas powerful-number sums in E940 use positive summands. It therefore does not imply that almost all admissible positive integers are sums of three positive cubes. Moreover, the congruence obstruction $a\equiv\pm4\pmod9$ is specific to three cubes and need not obstruct sums of arbitrary 3-powerful numbers.

The paper does not count arbitrary 3-powerful numbers, prove density zero or positive density unconditionally, or address any exponent $r\ge4$. Its relevance to E940 is concentrated in the conditional $r=3$ energy estimate and in the analytic framework—delta decomposition (2.10), the $\mathcal S_0/\mathcal S_1$ split, Ratios-type mean values, Proposition 8.1, and the bad-prime estimates of §9—that identifies what would be needed to turn that conditional counterexample into an unconditional one.