All Research

Abstract

We study sextic residue-character moments over the Eisenstein integers, with squarefree ideal columns of norm length D and element rows of norm at most H. The selected sixth-power rows yield a positive arithmetic form expressed through totient-weighted divisor means. A smooth Gram formula has an absolute O(D log²(2D)) remainder and gives a sufficient growing-test criterion with a sharp quadratic cutoff tail. At D = Hʳ with fixed r > 0, scale-dependent bounded coefficients can annihilate any prescribed fixed finite set of these means exactly while their selected sixth-power block retains energy of order H^(1/6)D. For the complete row space, we give an exact sixth-power decomposition and spectral criterion. Under a stated structured inverse-moment input, fixed-local coefficient modulation preserves the row-scale estimate; divisor representations with explicit norm costs and coefficient remainders give further sufficient transfers. The arguments specialize classical Selberg diagonalization and combine ideal counting, smooth lattice summation, Euler splitting and Hilbert-space inequalities.