A new convolution-type Bombieri-Vinogradov theorem with well-factorable weights improves the lower density for P^+(n)<P^+(n+1) from 0.280 to 0.296 and the upper density for P^+(p-1)>=p^c from (7/2)log(1/c) to a smaller S(c).
Ding,On a conjecture on shifted primes with large prime factors, Arch
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Convolution-type Bombieri-Vinogradov theorem with well-factorable Weights, and its applications
A new convolution-type Bombieri-Vinogradov theorem with well-factorable weights improves the lower density for P^+(n)<P^+(n+1) from 0.280 to 0.296 and the upper density for P^+(p-1)>=p^c from (7/2)log(1/c) to a smaller S(c).