Diagonal cubic forms and the large sieve

Abstract

Let $$ be the number of integral zeros $$ of $$. Works of Hooley and Heath-Brown imply $$, if one assumes automorphy and grand Riemann hypothesis for certain Hasse–Weil $$-functions. Assuming instead a natural large sieve inequality, we recover the same bound on $$. This is part of a more general statement, for diagonal cubic forms in $$ variables, where we allow approximations to Hasse–Weil $$-functions.