Arithmetic cutoff bounds and sharp tail capacity for local Nyman–Beurling certificates

Overview

Abstract

We determine the sharp tail capacity of anchored arithmetic trials in the whole-line Poisson norm at λ=α+iT, 1/2<α<1. After a prescribed exterior null is imposed, the complete norm has an M⁻¹ tail with a positive gcd-variance coefficient and an explicit period-free remainder. The normalized coefficient has exact infimum [12(ζ(2)ζ(2α)/|ζ(1+λ)|²−1)]⁻¹. Finite constrained capacities approach this limit at an explicit rate uniform in T. A dyadic factor imposes the null while preserving arbitrarily much projection gain at a paid support cost, yielding complete finite arithmetic lower certificates. Complementarily, an exact weighted divisor dual gives g_N(λ)≤K_α N^(2−2α)|ζ(λ)|² for all anchored complex coefficients supported on the integers at most N. The power is exact for the unrestricted head reconstruction norm. Montgomery's unconditional small-value theorem then gives a necessary support-cutoff growth rate for an all-height positive gain. For the same trials we also give a phase-preserving, finite-order expansion of the complete mean tail. Matching its two power kernels cancels the lower endpoint and improves the sharp fixed-trial mean order to M^(−α−1). Explicit corrected rows and full norm envelopes yield a constrained robust certificate with a unique scalar optimum. The summation and optimization machinery is classical; the arithmetic correction retains both nulls and the actual common head.

Keywords Nyman–Beurling · Dirichlet polynomials · Möbius inversion · gcd matrices · Poisson norms · Riemann zeta function