A matching lower bound for the sextic large sieve and structured prime moments

Abstract

We study the contrast between the large-sieve norm and the second moment of structured prime coefficients for sextic residue characters over the Eisenstein integers. For squarefree row and column ideals of norms at most A and B, we prove a lower bound of order (AB)^(-epsilon) times (A + B + A^(5/6) B^(1/3) + A^(1/3) B^(5/6)), matching de Faveri's upper bound throughout the full parameter range. In particular, the squared full-cutoff norm at balance is at least Q^(7/6 - epsilon). The proof uses the uniform unmarked character moments of OpenAI's September 2026 manuscript, a Gauss second moment uniform in moving prime exclusions, and a centered comparison that transports a squarefree-row signal between column scales. With the additional unmarked inverse moment and buffered zero detector from the same source, we derive a low-height zero-density estimate and bound the second moment of a fixed smooth prime sum by Q^((37 - 10 sqrt(6))/6 + epsilon). We also treat bounded convolutions of independently weighted large primes. An appendix retains the first version's unbalanced lower bound and its exponent 17/16, without using the additional plain or inverse moment estimates.