  Let $N(X)$ be the number of integral zeros $(x_1,\dots,x_6)\in [-X,X]^6$ of
$x_1^3+\dots+x_6^3$. 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 $m\geq 4$ variables, where we allow approximations to
Hasse--Weil $L$-functions.
