Friedlander–Granville–Montgomery conjecture on primes in arithmetic progressions

Let $x$ be real, let $q<x$ be a positive integer, let $a$ satisfy $(a,q)=1$, and let $\pi(x;q,a)$ count primes $p\leq x$ with $p\equiv a\pmod q$. Let $\varphi(q)$ denote Euler's totient function. Friedlander–Granville–Montgomery conjecture. For every $\varepsilon>0$, one has

$$\left|\pi(x;q,a)-\frac{\pi(x)}{\varphi(q)}\right|\ll_\varepsilon (x/q)^{1/2}x^\varepsilon.$$

In particular, the prime number theorem for arithmetic progressions holds uniformly for $q\ll x^{1-3\varepsilon}$. This corrected estimate is presented after Montgomery's stronger conjecture is described as overly optimistic, and it is then used to analyze the prime-generation algorithm.