On Hooley's delta function associated with characters (Q6583678)

The Hooley function is\N\[\N\Delta(n):=\max_{u\in {\mathbb R}} \sum_{\substack{d\mid n\\\Ne^u<d<e^{u+1}}} 1. \N\]\NA classical result of Hall and Tenenbaum is that the average of this function in the interval \([1,x]\) is \((\log x)^{o(1)}\). In the paper under review, the author looks at the two-dimensional variant of this function given by \N\[\N\Delta_3(n,f_1,f_2):=\sup_{\substack{(u_1,u_2)\in {\mathbb R}^2\\\N(v_1,v_2)\in [0,1]^2}} \left|\sum_{\substack{d_1d_2\mid n\\\Ne^{u_i}\le d_i\le e^{u_i+v_i}}} f_1(d_1)f_2(d_2)\right|, \N\]\Nwhere \(f_1,~f_2\) are arithmetic functions. The main result, which is Theorem 1.1, gives an upper bound on the second moment of \(\Delta_3(n,\chi_1,\chi_2)\), where \(\chi_1,~\chi_2\) are nonprincipal Dirichlet characters (and \(\chi_1{\overline{\chi_2}}\) is also nonprincipal). A (largely simplified) conclusion of the main result is that the average value of the second moment is bounded by \((\log x)^{\rho+o(1)}\) as \(x\to\infty\), where \(\rho={\sqrt{3}}/\pi-1/3\approx 0.218\ldots\). The long and technical proof uses ideas from the work of \textit{R. de la Bretèche} and \textit{G. Tenenbaum} [J. Lond. Math. Soc., II. Ser. 85, No. 3, 669--693 (2012; Zbl 1258.11086)] as well as a theorem of \textit{P. Shiu} [J. Reine Angew. Math. 313, 161--170 (1980; Zbl 0412.10030)]. This result is one of the main ingredients of the author's results on asymptotic counting of the number of ideals of fixed norm in a fixed cyclic extension of \({\mathbb Q}\) of degree \(3\) which is developed in the work [\textit{A. Lartaux}, Q. J. Math. 74, No. 2, 471--510 (2023; Zbl 1534.11116)].