Generalized Wirtinger inequalities, random matrix theory, and the zeros of the Riemann zeta-function. (Q1867448)

Let \(t_n>0\) denote the \(n\)-th zero of \(\zeta({1\over2}+it)\), arranged in increasing order and counted according to multiplicity, and define \(\Lambda=\limsup(t_{n+1}-t_n)/(2\pi/\log t_n)\). In an earlier paper [J. Number Theory 93, No. 2, 235--245 (2002; Zbl 0994.11030)] the author established that \(\Lambda\geq\sqrt{11/2}=2{\cdot}345207\ldots\,\). The method makes use of a Wirtinger type inequality, which gives a lower bound for a certain integral in terms of the fourth moment of a function involved. In the present paper the author shows that such an inequality can be generalised to higher moments, and with the sixth moment together with assumptions from Random Matrix Theory, he now derives the conditional bound \(\Lambda\geq\sqrt{7533/901}=2{\cdot}891489\ldots\,\).