Diagonal cubic forms and the large sieve

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 $N(X){\ll }_{\hat{I}}\mu X^{3+\hat{I}\mu }$, 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 $\ge 4$ variables, where we allow approximations to Hasse--Weil $L$-functions.

24 pages; removed appendices; improved exposition; accepted version