  It is a classic result of Selberg in the 1950's that $\theta_2 = 2/3$, where
$\theta_2$ is the level of distribution of the divisor function in arithmetic
progressions (defined more precisely below). Selberg applies this estimate,
together with his $\Lambda^2$ sieve, to prove weak forms of the binary Goldbach
and twin prime conjectures.
  In this note, we give an unconditional proof of Selberg's first result with
smooth weights for prime moduli, and put forth a hypothesis via subconvexity to
bootstrap to level $4/5$. Contingent outcomes of our proposal is an improvement
of Selberg's second result on approximations to twin primes and in lowering
bounds on gaps between primes.

Number Theory