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
 $\sum_{q\leq Q}\max_{y\leq x}\max_{1\leq a\leq q\atop(a,q)=1}\left|\psi(x;q,a)-% {x\over\phi(q)}\right|=O\left(x^{\frac{1}{2}}Q(\log x)^{5}\right),$
where $\phi(q)$ is Euler’s totient function and
 $\psi(x;q,a)=\sum_{n\leq x\atop n\equiv a\mod q}\Lambda(n),$
where $\Lambda(n)$ is the Mangoldt function.