Estimates for \(k\)-dimensional spherical summations of arithmetic functions of the GCD and LCM (Q6606729)

Let \(k\ge 1\). For a function \(F:\mathbb{N}^k\to\mathbb{C}\), in the paper under review, the authors study the following summation as \(x\to\infty\),\N\[\NS_F(x):=\sum_{\substack{n_1,\ldots,n_k\in \mathbb{N}\\\Nn_1^2+\cdots +n_k^2 \le x}} F(n_1,\ldots,n_k).\N\]\NThey assume that the associated multiple Dirichlet series\N\[\N\sum_{n_1,\ldots,n_k=1}^{\infty} \frac{(\mu\ast F)(n_1,\ldots,n_k)}{n_1^{z_1}\cdots n_k^{z_k}}\N\]\Nis absolutely convergent provided that \(z_j\in \mathbb{C}\) with \(\Re z_j\ge t\) for all \(1\le j\le k\), where \(0<t\le 1\) is a real number and \(\mu\) is the Möbius function. Letting \(V_k\) to be the volume of unit ball in \(\mathbb{R}^k\), and\N\[\NB_{F,k}: =\sum_{n_1,\ldots,n_k=1}^{\infty} \frac{(\mu\ast F)(n_1,\ldots,n_k)}{n_1\cdots n_k},\N\]\Nthey show that if \(t=1\), then \(S_F(x)\sim\frac{V_k}{2^k} B_{F,k}x^{k/2}\). For \(0<t<1\) they obtain a better approximation as follows\N\[\NS_F(x)=\frac{V_k}{2^k} B_{F,k} x^{k/2}+O(x^{(k-1+t)/2}).\N\]\NThe authors consider particular cases \(F=f(\gcd(n_1,\ldots,n_k))\) and \(F=f(\mathrm{lcm}(n_1,\ldots,n_k))\), for some specific arithmetic functions \(f\), including the well-known number theoretic functions \(\varphi, \lambda, \tau, \sigma, \omega, \Omega\) and \(\mu^2\).\N\NFor the entire collection see [Zbl 1530.11003].