The Bombieri-Vinogradov theorem, sometimes called Bombieri's theorem, states that for a positive real number $A$ , if $x^{\frac{1}{2}}\log^{-A}x\leq Q\leq x^{\frac{1}{2}}$ then $$\sum_{q\leq Q}\max_{y\leq x}\max_{1\le a\le q\atop (a,q) = 1}\left|\psi(x;q,a) - {x\over\phi(q)}\right| = O\left(x^{\frac{1}{2}}Q(\log x)^5\right),$$ where $\phi(q)$ is Euler's totient function and $$\psi(x;q,a)=\sum_{n\le x\atop n\equiv a\mod q}\Lambda(n),$$ where $\Lambda(n)$ is the Mangoldt function.