On the distribution of certain Hua sums (Q5947046)

The sums referred to in the title are NEWLINE\[NEWLINES(A,c)=\sum_{j=0}^{c}e(Aj^3/c),NEWLINE\]NEWLINE where \(A\) and \(c\) are positive integers. The paper shows that NEWLINE\[NEWLINE\sum_{c\leq X}S(A,c)=k(A)X^{4/3}+O_{A,\varepsilon}(X^{5/4+\varepsilon}).NEWLINE\]NEWLINE This contrasts with the situation for the analogous sums over \(\mathbb Z [\omega]\) (where \(\omega^2+\omega+1=0\)) for which \textit{R. Livné} and the author [Invent. Math. 148, No. 1, 79--116 (2002; Zbl 1039.11051)] have shown that an extra \(\log X\) factor may occur in the leading term.NEWLINENEWLINEThe key fact in the proof is that \(\sum_{c=1}^{\infty}S(A,c)c^{-s}\) has an analytic continuation to \(\Re(s)\geq 1\), and is regular except for a simple pole at \(s=4/3\). This is proved using the theory of metaplectic forms over \(\mathbb Q [\omega]\).