On certain exponential sums over primes (Q1024537)

Let \(f(x)\) be a real valued polynomial of degree \(k\geq 4\) and irrational leading coefficient \(\alpha\). Exponential sums of the form \[ S:=\sum_{p\leq N} (\log p) e(f(p)) \] have received a lot of interest. \textit{G. Harman} proved in [Mathematika 28, 249--254 (1981; Zbl 0465.10029)] that if \(q\) is the denominator of a convergent of \(\alpha\), then \[ S\ll N^{1+\varepsilon}\Bigl({{1}\over {q}}+{{1}\over {N^{1/2}}}+{{q}\over {N^k}}\Bigr)^{\gamma_1(k)},\quad \text{where}\quad \gamma_1(k)=4^{1-k}. \] Vinogradov showed that for large \(k\) one can take instead of \(\gamma_1(k)\) the quantity \((25k^2(2+\log k))^{-1}\). In the paper under review, the authors show that if \(\alpha\) is of type 1, namely if for all \(\varepsilon>0\) there are only finitely many integer solutions \((a,q)\) to the Diophantine inequality \(|\alpha-a/q|<q^{-2-\varepsilon}\), then \[ S\ll N^{1-\gamma^*(k)+\varepsilon},\quad \text{where}\quad \gamma^*(k)=(4(2^k+1))^{-1}. \] Since every \(\alpha\) of type \(1\) has a convergent whose denominator is in \([N^{1/2},N^{1/2+\varepsilon}]\) for large \(N\), Harman's result gives only that \[ S\ll N^{1-{{1}\over {2^{2k-1}}}+\varepsilon} \] so the result of the paper under review is better for such \(\alpha\)'s for all \(k\geq 4\). The current bound is also better than Vinogradov's result for \(k\in [4,11]\).