The prime number race and zeros of Dirichlet $L$-functions off the   critical line. III
  We show, for any $q\ge 3$ and distinct reduced residues $a,b \pmod q$, the existence of certain hypothetical sets of zeros of Dirichlet $L$-functions lying off the critical line implies that $\pi(x;q,a)<\pi(x;q,b)$ for a set of real $x$ of asymptotic density 1. 
Kevin Ford, Sergei Konyagin and Youness Lamzouri
math.NT