On exponential sums studied by Indlekofer and Kátai (Q1046940)

\textit{I. Kátai} [Acta Math. Hung. 112, No. 3, 221--225 (2006; Zbl 1150.11029)] conjectured that \(\Delta(\alpha,x)\to 0\) for irrational \(\alpha\). Here \(\Delta(\alpha,x)\) is defined by \[ \Delta(\alpha,x)=\frac{1}{\pi_2(x)}\max_{| X_p| \leq 1}\left| \sum_{\substack{ p_1p_2\leq x \\ p_1<p_2 }} X_{p_1}X_{p_2} e^{\alpha p_1p_2}\right| , \] where \(p_2(x)=\sum_{\substack{ p_1p_2 \leq \alpha \\ p-1<p_2}} 1\). The purpose of the present note is to prove this conjecture and disprove another one by \textit{K.-H. Indlekofer} and \textit{I. Kátai} [Acta Math. Hung. 118, No. 4, 313--318 (2008; Zbl 1164.11010)].