A lower explicit threshold for Chen's theorem under GRH

Abstract

Assume the Generalized Riemann Hypothesis (GRH). We show that every even integer N >= exp(exp(13.375)) can be written as N = p + P_2, where p is a prime and P_2 is either a prime or a product of two primes. The result is conditional on GRH and on Theorems 5, 6, 7 and Lemma 9 of Bordignon and Starichkova, Ramanujan J. 64 (2024), together with the auxiliary lemmas, Table 1 and display (6) that their Section 5.4 uses, at new parameter values; they obtain exp(exp(14)). It is not a new method: it is a re-choice of the parameters u and alpha in their argument, combined with the observation that Chen's theorem needs only that the Section 5.4 main-term combination be positive, not that it exceed 4*10^-3, because the inequality pi_2'(N) >= pi_2(N) - sqrt(N) - 1 from the proof of their Corollary 1 gives N = p + P_2 as soon as pi_2(N) > sqrt(N) + 1. The arithmetic of Section 5.4 at the new parameters and the parameter range inequalities are verified unconditionally in Lean 4.

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