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This problem was studied by Gerard Maze in Natural Density Distribution of Hermite Normal Forms of Integer Matrices (2010). One result:

The probability that an $n\times n$ integer matrix $A$ has an Hermite normal form (HNF) with the integers $d_1,d_2,\ldots,d_{n−1},d$ on the diagonal is given by $$\bigl(\zeta(n)\zeta(n-1)\cdots\zeta(2)d_1^n d_2^{n-1}\cdots d_{n-1}^2\bigr)^{-1}.$$

The expectation value and variance of a diagonal element have been calculated by Hu et al. in On random nonsingular Hermite Normal Form (2015, paywall). I quote their result:

Note that the distribution of the matrices is different in the two papers: Maze chooses the matrix elements uniformly in the interval $(-M,M)$ for large $B$, while Hu et al. choose the determinant uniformly from that interval.

This problem was studied by Gerard Maze in Natural Density Distribution of Hermite Normal Forms of Integer Matrices (2010). One result:

The probability that an $n\times n$ integer matrix $A$ has an Hermite normal form (HNF) with the integers $d_1,d_2,\ldots,d_{n−1},d$ on the diagonal is given by $$\bigl(\zeta(n)\zeta(n-1)\cdots\zeta(2)d_1^n d_2^{n-1}\cdots d_{n-1}^2\bigr)^{-1},$$ with $\zeta(n)$ the Riemann zeta function.

The expectation value and variance of a diagonal element have been calculated by Hu et al. in On random nonsingular Hermite Normal Form (2015, paywall). I quote their result:

Note that the distribution of the matrices is different in the two papers: Maze chooses the matrix elements uniformly in the interval $(-M,M)$ for large $M$, while Hu et al. choose the determinant uniformly from that interval.

This problem was studied by Gerard Maze in Natural Density Distribution of Hermite Normal Forms of Integer Matrices (2010). One result:

The probability that an $n\times n$ integer matrix $A$ has an Hermite normal form (HNF) with the integers $d_1,d_2,\ldots,d_{n−1},d$ on the diagonal is given by $$\bigl(\zeta(n)\zeta(n-1)\cdots\zeta(2)d_1^n d_2^{n-1}\cdots d_{n-1}^2\bigr)^{-1}.$$

The expectation value and variance of a diagonal element have been calculated by Hu et al. in On random nonsingular Hermite Normal Form (2015). I quote their result (since the paper is behind a paywall):

This problem was studied by Gerard Maze in Natural Density Distribution of Hermite Normal Forms of Integer Matrices (2010). One result:

The probability that an $n\times n$ integer matrix $A$ has an Hermite normal form (HNF) with the integers $d_1,d_2,\ldots,d_{n−1},d$ on the diagonal is given by $$\bigl(\zeta(n)\zeta(n-1)\cdots\zeta(2)d_1^n d_2^{n-1}\cdots d_{n-1}^2\bigr)^{-1}.$$

The expectation value and variance of a diagonal element have been calculated by Hu et al. in On random nonsingular Hermite Normal Form (2015, paywall). I quote their result:

Note that the distribution of the matrices is different in the two papers: Maze chooses the matrix elements uniformly in the interval $(-M,M)$ for large $B$, while Hu et al. choose the determinant uniformly from that interval.

This problem was studied by Gerad Maze in Natural Density Distribution of Hermite Normal Forms of Integer Matrices (2010). One result:

The probability that an $n\times n$ integer matrix $A$ has an Hermite normal form with the integers $d_1,d_2,\ldots,d_{n−1},d$ on the diagonal is given by $$\bigl(\zeta(n)\zeta(n-1)\cdots\zeta(2)d_1^n d_2^{n-1}\cdots d_{n-1}^2\bigr)^{-1}.$$

See also On random nonsingular Hermite Normal Form (paywall).

This problem was studied by Gerard Maze in Natural Density Distribution of Hermite Normal Forms of Integer Matrices (2010). One result:

The probability that an $n\times n$ integer matrix $A$ has an Hermite normal form (HNF) with the integers $d_1,d_2,\ldots,d_{n−1},d$ on the diagonal is given by $$\bigl(\zeta(n)\zeta(n-1)\cdots\zeta(2)d_1^n d_2^{n-1}\cdots d_{n-1}^2\bigr)^{-1}.$$

The expectation value and variance of a diagonal element have been calculated by Hu et al. in On random nonsingular Hermite Normal Form (2015). I quote their result (since the paper is behind a paywall):