The approximate functional equation of some Diophantine series (Q6109717)

Consider \(g:\mathbb R\to\mathbb R\) an odd \(1\)-periodic \(C^1\) function and \(f:\mathbb R\backslash\mathbb Z\to\mathbb R\) a \(1\)-periodic continuous function such that \(L=\lim_{x\to 0}(f(x)-\lambda/x)\) exists (and it is finite) for some \(\lambda\ne 0\). For \((w,\alpha)\in(\mathbb R^{+})^2\), introduce the Diophantine sum \[ \Phi_w(\alpha)=\sum_{m=1}^{\left\lfloor w\right\rfloor}\frac{g(m^2\alpha)}{m^2}f(m\alpha) , \] where \(\Phi_w(\alpha)\) is defined by continuity for \(\alpha=p/q\in\mathbb Q\) with \(q\le w\). In [\textit{T. Rivoal} and \textit{J. Roques}, Unif. Distrib. Theory 8, No. 1, 97--119 (2013; Zbl 1349.11102)] it is proved that when \(g(x)=\sin(2\pi x)\) and \(f(x)=\cot(\pi x)\), \(\Phi(\alpha)\) satisfies an approximate functional equation for a positive integer \(N\). In this paper it is shown that this approximate functional equation also holds with the general definition of \(\Phi(N)\) and that it can be deduced from a mainly combinatorial argument not depending on the choice of \(f\) and \(g\). The continuity of the limiting function for \(\alpha>0\) and its continuous extension to \([0,\infty)\) are also proved, settling in particular the conjecture posed in [\textit{T. Rivoal} and \textit{J. Roques}, Unif. Distrib. Theory 8, No. 1, 97--119 (2013; Zbl 1349.11102)]. By applying a general result of \textit{M. Balazard} and \textit{B. Martin} [Aequationes Math. 93, No. 3, 563--585 (2019; Zbl 1435.11126)] for certain approximate functional equations, the convergence points of \(\Phi(N)\) are completely characterized when \(N\to\infty\).