Laguerre inequalities and complete monotonicity for the Riemann Xi-function and the partition function

Abstract:

In this paper, we find some conditions under which a sequence $\{\alpha (n)\}$ will satisfy the Laguerre inequality of any order asymptotically. Using this method, we prove that for any $r$ and some constant $c$, the Maclaurin coefficients $\gamma (n)$ of the Riemann Xi-function satisfy the Laguerre inequality of order $r$ when $n>cr^3$, which provides a necessary condition for the Riemann hypothesis. We also prove that the partition function satisfies the Laguerre inequality of order $r\geq 5$ when $n\geq 6r^4$. As a consequence, it gives an affirmative answer to Wagner’s conjecture on the threshold for the Laguerre inequalities of order no more than $10$ for the partition function. Moreover, motivated by the study of Craven and Csordas on the complete monotonicity of the Maclaurin coefficients of entire functions in Laguerre-Pólya class, we consider the complete monotonicity of the sequences $\{\alpha (n)\}$. We give the criteria for the asymptotically complete monotonicity of the sequence $\{\alpha (n)\}$ and $\{\log \alpha (n)\}$, respectively. With this criteria, we show that $(-1)^r \Delta ^r \gamma (n)>0$ for $n>ce^{r^3}$ and $(-1)^{r-1} \Delta ^r \log \gamma (n)>0$ for $n>cr^2$. Furthermore, we propose some open problems.