Laguerre inequalities and complete monotonicity for the Riemann Xi-function and the partition function (Q6582244)

In this paper under review, the authors deal with a general family of sequences that includes the Maclaurin coefficients of the Riemann Xi-function \(\gamma(n)\) and the partition function \(p(n)\). More generally, they prove under some conditions that a sequence \(\lbrace\alpha(n)\rbrace\) satisfies the Laguerre inequalities of any order \(r\) when \(n\) is larger than a computable bound \(N(r)\). In particular, they give explicit expression of \(N(r)\) for \(\gamma(n)\) and \(p(n)\). Finally, they give the criteria for the asymptotically complete monotonicity of a sequence \(\lbrace\alpha(n)\rbrace\) and \(\lbrace \log \alpha(n)\rbrace\), respectively. Using this criteria, it is shown that \((-1)^{r-1} \Delta^r \log \gamma (n)>0\) for \(n>cr^2\) and \((-1)^r \Delta^r \gamma (n)>0\) for \(n>ce^{r^3}\).