David Dummit and Hershy Kisilevsky observed from calculation that the Legendre symbols (p/q) and (q/p) are unequal for rather more than a quarter of the pairs of odd primes p and q with pq<=x, during some calculations. In fact almost 30% of the pq's up to a million satisfy p=q=3 mod 4. Together we found that this is no accident and that the bias up to x is roughly 1 +1/3(loglog x1). This is a much stronger bias than the traditional "prime race" problem. When doing the math one finds that this problems about pq's is equivalent to the prime race problem, for primes =3 mod 4 versus those =1 mod 4, in which we weight each prime by its reciprocal.
Primes in Intervals of Bounded Length
Basics of binary quadratic forms and Gauss composition
Don't be seduced by the zeros!
Different approaches to the distribution of primes
The Princeton Companion to Mathematics: Analytic number theory
Prime number patterns (2009 Ford Prize)
It is easy to determine whether a given integer is prime (2008 Chauvenet Prize)
Prime number races (2007 Ford Prize)
It's as easy as abc
Zaphod Beeblebrox's brain and the fiftyninth row of Pascal's triangle (1995 Hasse Prize)
