Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 27 / 4 / b

Put $E_q={\max }_{(a,q)=1}|\pi (x;q,a)-\mathrm{Li}(x)/\varphi (q)|$, with $\mathrm{Li}$ the offset logarithmic integral function. The permitted standard sieve estimate can be taken as the Brun–Titchmarsh theorem:For $q\le x^{0.99}$, $\log (x/q)\ge 0.01\log x$. Also $\mathrm{Li}(x)\ll x/\log x$. Thus both terms defining $E_q$ are $O(x/(\varphi (q)\log x))$, uniformly over the permitted moduli. Apply the Cauchy-Schwarz inequality in the formThis is the weighted arithmetic-progression error bound. The prime omega function and Euler totient function weights can therefore be handled using an unweighted mean error and a separate positive sum.