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For fixed $b,c$, $$|\sum_{bcl \leq x} e(\alpha \sqrt{bcl})| \ll x^{1/2} (bc)^{-1} \alpha^{-1}+ \alpha^2 bc$$ follows easily from lemma 1. The sum is over $l \leq x/bc$, and the "$\alpha$" in lemma 1 will be replaced $\alpha \sqrt{bc}$.
