On the error term concerning the number of subgroups of the groups \(\mathbb{Z}_m \times \mathbb{Z}_n\) with \(mn \leq x\)
For positive integers \(m,n\), let \(s(m,n)\) and \(c(m,n)\) denote respectively, the number of subgroups and the number of cyclic subgroups of \(\mathbb{Z}_m\times \mathbb{Z}_n\). For \(x\geq 1\), define \(D_s(x):=\sum_{mn \leq x}s(m,n)\) and \(D_c(x):=\sum_{mn \leq x}c(m,n)\). The authors prove the following asymptotic formulae by using the method of exponential sums: \[ D_s(x)=xP_4(\log x)+ O(x^{\frac 23}\log^6x),\quad D_c(x)=xR_4(\log x)+ O(x^{\frac 23}\log^6x) \] where \(P_4(u)\) and \(R_4(u)\) are polynomials in \(u\) of degree 4 with leading coefficients \(\frac 1{8\pi^2}\) and \(\frac 3{4\pi^4}\) respectively. Now consider the error terms of the asymptotic formulae of \(D_s(x),D_c(x)\) given by \(\Delta_s(x)=D_s(x)-xP_4(\log x)\), \(\Delta_c(x)=D_c(x)-xR_4(\log x)\). The authors prove the following mean square estimates: for any real \(T\geq 10\), \[ \int_{1}^{T} \Delta_s^2(x)\,dx \ll T^{\frac {77}{36}+\varepsilon},\quad \int_{1}^{T} \Delta_c^2(x)\,dx \ll T^{\frac {77}{36}+\varepsilon}. \]
scientific article; zbMATH DE number 7226991