Higher order Turán inequalities for the Riemann $\xi$-function

Abstract:

The simplest necessary conditions for an entire function \center $\displaystyle \psi (x) = \sum _{k=0}^{\infty } \gamma _{k} \frac {x^{k}}{k!}$ \endcenter to be in the Laguerre-Pólya class are the Turán inequalities $\gamma _k^2 - \gamma _{k+1} \gamma _{k-1} \geq 0$. These are in fact necessary and sufficient conditions for the second degree generalized Jensen polynomials associated with $\psi$ to be hyperbolic. The higher order Turán inequalities $4(\gamma _n^2-\gamma _{n-1}\gamma _{n+1})(\gamma _{n+1}^2 - \gamma _n\gamma _{n+2})-(\gamma _n\gamma _{n+1}-\gamma _{n-1}\gamma _{n+2})^2\geq 0$ are also necessary conditions for a function of the above form to belong to the Laguerre-Pólya class. In fact, these two sets of inequalities guarantee that the third degree generalized Jensen polynomials are hyperbolic.

Pólya conjectured in 1927 and Csordas, Norfolk and Varga proved in 1986 that the Turán inequalities hold for the coefficients of the Riemann $\xi$-function. In this short paper, we prove that the higher order Turán inequalities also hold for the $\xi$-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three.