Sums of cubes and the Ratios Conjectures

TL;DR

This paper improves bounds on the number of solutions to a specific cubic equation, removing epsilon factors under certain hypotheses, and connects these results to conjectures about sums of three cubes and prime distributions.

Contribution

It removes the epsilon in bounds for solutions to a cubic equation by assuming standard number-theoretic hypotheses, including Ratios and Square-free Sieve Conjectures.

Findings

Conditional bounds on solutions to $x_1^3+\

Almost all integers not congruent to ±4 mod 9 are sums of three cubes.

Under stronger hypotheses, power-saving asymptotics for the number of solutions and primes are obtained.

Abstract

Works of Hooley and Heath-Brown imply a near-optimal bound on the number $N$ of integral solutions to $x_1^3+\dots+x_6^3 = 0$ in expanding regions, conditional on automorphy and GRH for certain Hasse--Weil $L$-functions; for regions of diameter $X\ge 1$, the bound takes the form $N\le C(\varepsilon) X^{3+\varepsilon}$ ($\varepsilon>0$). We attribute the $\varepsilon$ to several subtly interacting proof factors; we then remove the $\varepsilon$ assuming some standard number-theoretic hypotheses, mainly featuring the Ratios and Square-free Sieve Conjectures. In fact, our softest hypotheses imply conjectures of Hooley and Manin on $N$, and show that almost all integers $a\not\equiv \pm 4 \bmod{9}$ are sums of three cubes. Our fullest hypotheses are capable of proving power-saving asymptotics for $N$, and producing almost all primes $p\not\equiv \pm 4 \bmod{9}$.