Zeta values, random matrix theory and Euler-MacLaurin summation

Let $\alpha$ be a real number greater than one and $\beta$ a positive real number. We prove that $\left(\zeta(\alpha + \beta n)\right)_{n\in\mathbb{N}}$ arise as moments of a positive definite Borel measure and construct the corresponding matrix theory. We determine its asymptotic behavior and show its relation to Euler-MacLaurin summation.