When his record company told avant-garde musician Anthony Moore that his work was not commercial enough he wasn't happy. Or rather he was Slapp Happy, the name he gave to his new band with their deliberately commercial sound. The band became less commercial subsequently but not before songs such as 'Blue Flower' bloomed. Dagmar Krause provides the distinctive voice.
We show the primes have level of distribution 66/107 using triply well-factorable weights. This gives the highest level of distribution for primes in any setting, improving on the prior record level 3/5 of Maynard. We also extend this level to 5/8, assuming Selberg's eigenvalue conjecture. As a result, we obtain new upper bounds for twin primes and for Goldbach representations of even numbers $a$. For the Goldbach problem, this is the first use of a level of distribution beyond the 'square-root barrier', and leads to the greatest improvement on the problem since Bombieri--Davenport from 1966.
