Random systems of equations in free abelian groups

We study the solvability of random systems of equations on the free abelian group $\mathbb Z^m$ of rank $m$. Denote by $\operatorname{SAT}(\mathbb Z^m,k,n)$ and $\operatorname{SAT}_{\mathbb Q^m}(\mathbb Z^m,k,n)$ the sets of all systems of $n$ equations of $k$ unknowns in $\mathbb Z^m$ satisfiable in $\mathbb Z^m$ and $\mathbb Q^m$ respectively. We prove that the asymptotic density $\rho(\operatorname{SAT}_{\mathbb Q^m}(\mathbb Z^m,k,n))$ of the set $\operatorname{SAT}_{\mathbb Q^m}(\mathbb Z^m,k,n)$ equals 1 for $n\le k$ and 0 for $n>k$. As regards, $\operatorname{SAT}(\mathbb Z^m,k,n)$ for $n<k$, some new estimates are obtained for the lower and upper asymptotic densities and it is proved that they lie between $\left(\prod^k_{j=k-n+1}\zeta(j)\right)^{-1}$ and $\left(\frac{\zeta(k+m)}{\zeta(k)}\right)^n$, where $\zeta(s)$ is the Riemann zeta function. For $n\le k$, a connection is established between the asymptotic density of $\operatorname{SAT}(\mathbb Z^m,k,n)$ and the sums of inverse greater divisors over matrices of full rank. Starting from this result, we make a conjecture about the asymptotic density of $\operatorname{SAT}(\mathbb Z^m,n,n)$. We prove that $\rho(\operatorname{SAT}(\mathbb Z^m,k,n))=0$ for $n>k$.