On the minimum trace norm of (0,1)-matrices

The trace norm of a matrix is the sum of its singular values. This paper presents results on the minimum trace norm $\psi_{n}\left( m\right)$ of $\left( 0,1\right)$-matrices of size $n\times n$ with exactly $m$ ones. It is shown that: (1) if $n\geq2$ and $n<m\leq2n,$ then $\psi_{n}\left( m\right) \leq \sqrt{m+\sqrt{2\left( m-1\right) }}$ , with equality if and only if $m$ is a prime; (2) if $n\geq4$ and $2n<m\leq3n,$ then $\psi_{n}\left( m\right) \leq \sqrt{m+2\sqrt{2\left\lfloor m/3\right\rfloor }}$ , with equality if and only if $m$ is a prime or a double of a prime; (3) if $3n<m\leq4n,$ then $\psi_{n}\left( m\right) \leq\sqrt{m+2\sqrt{m-2}}$ , with equality if and only if there is an integer $k\geq1$ such that $m=12k\pm2$ and $4k\pm1,6k\pm1,12k\pm1$ are primes.