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 $Ξ_6(A,B)$ 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 $0<δ<3/11$ and every $\varepsilon>0$, we prove $Ξ_6(A,B)\gg_{δ,\varepsilon}(AB)^{-\varepsilon}A^{5/6}B^{1/3}$ for $B^{16/11+δ}\le A\le B^{2-δ}$. Consequently, the full-cutoff norm satisfies $Ξ_6(X,X)\gg_\varepsilon X^{17/16-\varepsilon}$. 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 Möbius 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.