Quadratic primes and discrepancy in short aritmetic progressions

Abstract

It is shown that the asymptotic size of the mean discrepancy function for primes in arithmetic progressions with unusually large square common difference furnishes a necessary and sufficient condition for the infinitude of quadratic primes. In particular, the size of the mean discrepancy over very short progressions of primes with square difference is strong enough to dictate the number of primes of the form 4n2 + 1 as predicted by the Bateman-Horn conjecture. Conversely, assuming the Bateman-Horn conjecture, we show how the mean discrepancy function behaves as expected by exhibiting a strong asymptotic negative tendency. By estimating the mean discrepancy from above with the help of Selberg's upper bound sieve, and keeping track of the actual values of the involved constants together with a fair amount of optimization, we are able to confirm this negative tendency unconditionally over certain ranges of the parameters, thereby forming evidence supporting the truth of the Bateman-Horn conjecture. Our confirmed upper bounds reveal that in a sense one comes within a hair's breadth away from settling the infinitude of quadratic primes. Moreover, by developing a uniform version of Selberg's sieve based on a symmetric Mertens type inequality, we extend the mean discrepancy estimates to slightly longer progressions using delicate analytic properties of the arithmetic of imaginary quadratic fields. Lastly, we obtain a Brun-Titchmarsh type inequality holding uniformly for most of the reduced residue classes of primes belonging to a progression having a very large square common difference with the help of an asymptotic formula on the average value of Chebyshev's psi-function over such moduli.