  The primes or prime polynomials (over finite fields) are supposed to be
distributed `irregularly' , despite nice asymptotic or average behavior. We
provide some conjectures/guesses/hypotheses with `evidence' of surprising
symmetries in prime distribution. At least when the characteristic is $2$, we
provide conjectural rationality and characterization of vanishing for families
of interesting infinite sums over irreducible polynomials over finite fields.
The cancellations responsible do not happen degree by degree or even for degree
bounds for primes or prime powers, so rather than finite fields being
responsible, interaction between all finite field extensions seems to be
playing a role and thus suggests some interesting symmetries in the
distribution of prime polynomials. Primes are subtle, so whether there is
actual vanishing of these sums indicating surprising symmetry (as guessed
optimistically), or whether these sums just have surprisingly large valuations
indicating only some small degree cancellation phenomena of low level
approximation symmetry (as feared sometimes pessimistically!), remains to be
seen. In either case, the phenomena begs for an explanation. (Version 2: A
beautiful explanation by David Speyer and important updates are added at the
end after the first version.)
