A sharp lower bound for the sextic large sieve

Abstract

We establish a matching lower bound for the sextic large sieve over the Eisenstein integers in an unbalanced range, with both indices restricted to squarefree ideals. Let $\Xi_6(A,B)$ denote the squared operator norm of the sextic residue-character matrix with row and column norms at most $A$ and $B$, respectively, outside a fixed finite set of primes containing those above $6$. For every fixed $0<\delta<3/11$ and every $\varepsilon>0$, we prove $\Xi_6(A,B)\gg_{\delta,\varepsilon}(AB)^{-\varepsilon}A^{5/6}B^{1/3}$ for $B^{16/11+\delta}\le A\le B^{2-\delta}$. Consequently, the full-cutoff norm satisfies $\Xi_6(X,X)\gg_\varepsilon X^{17/16-\varepsilon}$. The main difficulty is to preserve the large second moment supplied by Gauss sums when the column moduli are restricted to be squarefree. We prove a second-moment formula uniform in the prime exclusions and character twists arising from M\"obius inversion, and use it to show that the squarefree projection retains a positive proportion of the original moment. A separate argument combining cubic and sextic large-sieve bounds then transfers the lower bound to squarefree rows.