Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 150 / 4 / e / Solution

Assume $\beta$ exists and choose a reduced residue class $a$ with ${\chi }_1(a)=-1$; such a class exists because ${\chi }_1$ is nonprincipal. Fix a sufficiently large constant $A=A(ϵ)$ and takeThen $x\ge q^2$ for large $q$. Multiplying the formula from part (d) by $\phi (q)/x$ givesChoose $A$ so large that the error is at most $ϵ/4$ for all sufficiently large $q$. The assumed upper bound then impliesSince $\beta <1$, this givesTaking logarithms,Therefore, with a positive constant depending only on $ϵ$,