Trace and Frobenius Euler Products for a Shifted Divisor Matrix

For the divisors greater than one of an integer \(a\), this preprint studies a weighted negative inverse of the shifted power GCD matrix. It derives exact Euler-product formulas for the trace of the square and the squared Frobenius norm of the resulting operator. The proof uses a prime-power tridiagonal factorization and computes the distinct row and column corrections caused by deleting the divisor \(1\). The formulas hold for every integer \(a\ge2\) and every positive integer exponent \(k\)