MathlibNt.SieveTheory.LinearSieve

Linear sieve / Jurkat-Richert estimates

The Jurkat-Richert theorem (1965) is a central tool for the lower bound on W(N) in Chen's theorem. It expresses upper and lower sieve bounds in terms of functions F(s) and f(s).

The differential-delay framework motivating the legacy outline uses:

• F(s) = 2e^γ / s for 2 ≤ s ≤ 4;
• f(s) = 0 for s ≤ 3;
• (s·F(s))' = f(s-1) for s ≥ 4;
• (s·f(s))' = F(s-1) for s ≥ 3, where γ is the Euler-Mascheroni constant.

These legacy conventions are not a specification of the canonical dimension-one sieve functions. In particular, sieveFunctionF and sieveFunctionf in the imported SieveApplications module contain placeholder branches and are used only in fixed-parameter remainder interfaces. They do not establish the classical uniform estimates or Chen's numerical constants. The finite lower-Möbius and generic Rosser density interfaces are developed separately in the imported modules.

References:

• Jurkat & Richert (1965), Acta Arith. 11, 217-240
• Halberstam & Richert, "Sieve Methods" (1974), Ch. 8
• Liu, Z. (2022), arXiv:2203.07871, §III