Diagonal cubic forms and the large sieve

TL;DR

This paper links the distribution of zeros of diagonal cubic forms to a natural large-sieve inequality. By implementing a delta-method framework and carefully partitioning contributions into smooth, central, and singular hyperplane sections, it shows that a suitably strong large-sieve hypothesis implies bounds on the number of integral zeros $N_F(X)$ of the form $N_F(X)circ X^{3(m-2)/4+oldsymbol{amma}}$, in particular recovering the near-optimal $X^{3+oldsymbol{amma}}$ bound for $m=6$. The authors develop a robust machinery: a delta-method with S_c(n), I_c(n), and Dirichlet-series approximations Ψ of Φ, a conversion from standard to non-standard coefficient families, and a detailed analysis of smooth, central, and singular hyperplane contributions. They also provide concrete instances of Ψ (via products of local factors and L-functions) for which the required large-sieve bounds hold, thereby tying automorphic input to unconditional-looking, large-sieve–driven estimates. The work broadens the conditional landscape for diagonal cubic forms, offering a unified path from large-sieve inequalities to sharp bounds across multiple variables.

Abstract

Let $N(X)$ be the number of integral zeros $(x_1,\dots,x_6)\in [-X,X]^6$ of $\sum_{1\le i\le 6} x_i^3$. Works of Hooley and Heath-Brown imply $N(X)\ll_εX^{3+ε}$, if one assumes automorphy and GRH for certain Hasse--Weil $L$-functions. Assuming instead a natural large sieve inequality, we recover the same bound on $N(X)$. This is part of a more general statement, for diagonal cubic forms in $\geq 4$ variables, where we allow approximations to Hasse--Weil $L$-functions.

Theorems & Definitions (56)

• Theorem 1.1
• Definition 2.2
• Theorem 2.3
• Definition 2.5
• Theorem 2.7
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3