The “trivial” bound in this context is $|{\mathcal{N}}r|^{\frac{1}{4}+{\varepsilon}}$ on the right hand side, while the Ramanujan conjecture (implied by the Lindelöf hypothesis for twisted $L$-functions) states the bound $|{\mathcal{N}}r|^{{\varepsilon}}$.

One particular situation where such bounds are needed, are asymptotic formulae for the number of representations of totally positive integers by ternary quadratic forms, see [@Bl] for an overview of this topic over ${\mathbb{Q}}$. Hilbert’s eleventh problem asks more generally which integers are integrally represented by a given $n$-ary quadratic form $Q$ over a number field $K$. If $Q$ is a binary form, it corresponds to some element in the class group of a quadratic extension of $K$ (see [@Cox] for a nice account over ${\mathbb{Q}}$). If $Q$ is indefinite at some archimedean place, Siegel [@Si] for $n {\geqslant}4$ and Kneser [@Kn] and Hsia [@H] for $n=3$ proved a local-to-global principle, so Siegel’s mass formula [@Si1] tells us exactly which integers are represented by $Q$. If $Q$ is positive definite at every archimedean place and $n {\geqslant}4$, again Siegel’s mass formula [@Si1] and bounds for Fourier coefficients of Hilbert modular forms give a complete answer (some care has to be taken in the case $n=4$). The only remaining case of $Q$ 