Hardy and Littlewood proved the three prime theorem under the GRH(generalized Riemann hypothesis) in an old paper: Some problems of `Partitio numerorum'; III: On the expression of a number as a sum of primes. Acta Math. 44 (1923), no. 1, 1–70.
The proof is long with old notations. Is there a modern reference for their proof? I mean, modern people should be able to simplify their proof to make it shorter than Vinogradov's unconditional proof given in Nathanson's Additive Number Theory textbook. My main focus is: how does GRH help in simplifying Vinogradov's proof? I heard that with GRH, the estimate for minor arc is much simpler but I don't know the details.