Higher order turán inequalities for the Riemann ξ-function

Dimitrov, Dimitar Kolev [UNESP]; Lucas, Fábio R.

Higher order turán inequalities for the Riemann ξ-function

Arquivos

2-s2.0-79951846250.pdf (482.42 KB)

Data

2011-03-01

Autores

Dimitrov, Dimitar Kolev

Lucas, Fábio R.

Editor

Amer Mathematical Soc

Tipo

Artigo

Direito de acesso

Acesso aberto

Resumo

The simplest necessary conditions for an entire function ψ(x) =∞ ∑ k=0 γk xk/k! to be in the Laguerre-Pólya class are the Turán inequalities γ2 k- γk+1γk-1 ≥ 0. These are in fact necessary and sufficient conditions for the second degree generalized Jensen polynomials associated with ψ to be hyperbolic. The higher order Turán inequalities 4(γ2 n - γn-1γn+1)(γ2n +1 - γnγn+2) - (γnγn+1 - γn-1γn+2) 2 ≥ 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 ψ-function. In this short paper, we prove that the higher order Turán inequalities also hold for the ψ-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three. © 2010 American Mathematical Society.