The relative entropy of primes in arithmetic progressions is really small

Abstract

Fix a modulus $q$. One would expect the number of primes in each invertible residue class mod $q$ to be multinomially distributed, i.e. for each $p \,\mathrm{mod}\, q$ to behave like an independent random variable uniform on $(\mathbb{Z}/q\mathbb{Z})^\times$. Using techniques from data science, we discover overwhelming evidence to the contrary: primes are much more uniformly distributed than iid uniform random variables. This phenomenon was previously unknown, and there is no clear theoretical explanation for it. To demonstrate that our test statistic of choice, the KL divergence, is indeed extreme, we prove new bounds for the left tail of the relative entropy of the uniform multinomial using the method of types.