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.