A random anchored sieve ensemble: exact expectation and variance theorems with a complete decomposition of the variance

Abstract

We introduce a probability model for sieve problems in which the sieve con straints, rather than the integers, are randomized: for every odd prime p ≤ z an anchor cp ∈ Z/pZ is drawn independently and uniformly, and we count the members of a fixed survivor set S0 that avoid every anchored residue class. The classical Goldbach sieve is the deterministic specialization cp = N mod p; Cramér-type models randomize the integers, whereas the present ensemble randomizes the sieve itself. For this model we prove a complete theorem system at both the expectation and the variance level. (i) E[W] = MQ ex actly (Theorem 3.1). (ii) An anchor compensation identity (Theorem 3.2): the ratio of the conditional expectation to the sieve main term is the same constant M/(LQ) for every anchor zero-set Z = {p : cp = 0}; equivalently, the Hardy–Littlewood singular series is absorbed exactly at the expectation level, and no anchor is exceptional at the expectation level. (iii) Two exact variance formulas (Theorems 4.1 and 4.2), the second showing that the main terms cancel identically because P2 = C2(z) Q2 is an exact Euler-product identity. (iv) A law-of-total-variance split (Theorem 5.1) with the between class part in closed form, VarZE[W | Z] = (MQ) 2