The asymptotic distribution of coefficients of the Dedekind zeta-function over a sparse sequence (Q6545058)

Let \(K\) be an algebraic number field of degree \(n \geqslant 2\), and let \(\zeta_K(s)\) be the Dedekind zeta function attached to \(K\) with residue at \(s=1\) denoted by \(\kappa_K\). If \(r_K(m)\) is the \(m\)th coefficient of \(\zeta_K(s)\), then the Ideal Theorem is the study of the error term \(\Delta_K (x)\) defined by \(\Delta_K (x) := \sum_{m \leqslant x} r_K(m) - \kappa_K x\) for \(x > 0\). In the paper under review, the authors investigate the following related sum \N\[\N\sum_{m = m_1^2+m_2^2 \leqslant x} r_{K_3}(m)^\ell\N\]\Nin the case where \(K=K_3\) is a non-normal cubic number field, and where \(m_1,m_2 \in \mathbb{Z}\) and \(\ell \in \mathbb{Z}_{\geqslant 2}\). The case \(\ell = 1\) has been considered in [\textit{Z. Yang}, Front. Math. China 12, No. 4, 981--992 (2017; Zbl 1427.11122)], and the present work improves on [\textit{G. Hu} and \textit{K. Wang}, Front. Math. China 15, No. 1, 57--67 (2020; Zbl 1466.110087)]. The method of proof is essentially the same as in these articles, but the improvements come from different novelties: in particular, the authors use in a crucial way the recent breakthrough of \textit{J. Newton} and \textit{J. A. Thorne} [Publ. Math., Inst. Hautes Étud. Sci. 134, 117--152 (2021; Zbl 1503.11086)] stating that, if \(\pi\) is a non-CM regular algebraic cuspidal automorphic representation of \(\textrm{GL}_{2} \left( \mathbb{A}_{\mathbb{Q}}\right) \), then \(\textrm{sym}^j \, \pi\) is a regular algebraic automorphic cuspidal representation of \(\textrm{GL}_{j+1} \left( \mathbb{A}_{\mathbb{Q}}\right) \) for all \(j \in \mathbb{Z}_{\geqslant 1}\). This allows the authors to decompose the Dirichlet series of the arithmetic function \(r_{K_3}(m)^\ell \left( \chi_4 \star \mathbf{1}\right) (m)\) into a product of more manageable \(L\)-functions, where \(\chi_4\) is the primitive Dirichlet character modulo \(4\), which in turn allows them to take advantage of better individual and mean value convexity bounds of the Riemann zeta-function and the symmetric square \(L\)-function.