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 $N(X)\ll_\epsilon X^{ 3+\epsilon}$, 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.
Categories: math.NT