Higher order Turán inequalities for the Riemann $\xi$-function
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- by Dimitar K. Dimitrov and Fábio R. Lucas PDF
- Proc. Amer. Math. Soc. 139 (2011), 1013-1022 Request permission
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.
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Additional Information
- Dimitar K. Dimitrov
- Affiliation: Departamento de Ciências de Computação e Estatística, IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil
- MR Author ID: 308699
- Email: dimitrov@ibilce.unesp.br
- Fábio R. Lucas
- Affiliation: Departamento de matemática Aplicada, IMECC, UNICAMP, 13083-859 Campinas, SP, Brazil
- Email: fabio25jk@hotmail.com
- Received by editor(s): January 8, 2010
- Received by editor(s) in revised form: March 5, 2010, and March 28, 2010
- Published electronically: September 15, 2010
- Additional Notes: Research supported by the Brazilian Science Foundations FAPESP under Grants 03/01874-2 and 06/60420-0, CNPq under Grant 305622/2009-9, and CAPES under Grant CAPES/DGU-160.
- Communicated by: Walter Van Assche
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 1013-1022
- MSC (2010): Primary 33E20, 11M06; Secondary 26D07
- DOI: https://doi.org/10.1090/S0002-9939-2010-10515-4
- MathSciNet review: 2745652