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In Waring's problem, we have Hua's estimate

$$S(a,b,q) = \sum_{x=1}^q e^{2\pi i (ax^k + bx)/q)} \ll q^{1/2+\epsilon} \gcd(b,q),$$ where $(a,q)=1$.

?Do you know a similar upper bound for the sum $$T(a,b,q) = \sum_{\substack{1 \le x \le q\\ (x,q)=1}} e^{2\pi i (ax^k + bx)/q)}$$ where $(a,q)=1$, which appears in Waring-Goldbach problem.

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This sum is explicitly studied in the recent paper by Anderson-Cook-Hughes-Kumchev https://arxiv.org/abs/1703.02713 , they refer to a an estimate of Shparlinski obtaining a bound of $O(q^{1/2+\epsilon})$ in the case $gcd(a,b,q)=1$, I don't know if this bound is sharp.

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