Large spaces between the zeros of the Riemann zeta-function and random matrix theory
Let \(\{t_n\}\) be the sequence of positive real zeros of the function \(\zeta(\frac12+it)\), counted according to multiplicity and arranged in non-decreasing order. The author investigates the large gaps between those zeros, namely \[ \Lambda:=\limsup\frac{t_{n+1}-t_{n}}{(2\pi/\log t_n)}. \] In [J. Number Theory 93, No. 2, 235--245 (2002; Zbl 0994.11030)] he proved that \[ \Lambda\geq\sqrt{\frac{11}{2}}=2.345207\dots. \] The author's method depends on the moment formulae \[ \int_0^T Z(t)^{2k-2h} Z'(t)^{2h}\,dt\sim C(h,k)T(\log T)^{k^2+2h},\tag{\(*\)} \] where \(Z(t)\) is Hardy's \(Z\)-function and this formulae are known in each of the cases \(h, k\in \mathbb{N}\), \(0\leq h\leq k\leq 2\). The above bound for \(\Lambda\) was obtained using (\(*\)) with \(k=2\) and \(h=0,1,2\). Note, that applied with \(k=1\), \(h=0,1\) the method yields \(\Lambda\geq\sqrt{3}\). C. P. Hughes showed that the Random Matrix Theory (RMT) predicts (\(*\)) in the cases \(1\leq h\leq k\). Let us denote the lower bound obtainable from (\(*\)) for a particular \(k\in\mathbb{N}\), employing every \(h\), by \(\Lambda(k)\). Thus \(\Lambda(1)\geq\sqrt{3}\), \(\Lambda(2)\geq\sqrt{(11/2)}\). In [J. Number Theory 97, No. 2, 397--409 (2002; Zbl 1066.11037)] the author obtained that \(\Lambda(3)\geq\sqrt{(7533/901)}=2.891\dots\). In this paper the author simplifies his method and proves that \[ \Lambda(4)\geq\sqrt{\frac{138046755}{11996228}}=3.392272\dots, \] \[ \Lambda(5)\geq\sqrt{\frac{10103742432235595}{678525561860941}}=3.858851\dots \quad \text{and}\quad\Lambda(6)\geq4.2981467\dots, \] where the last quantity is also a quadratic surd.
scientific article; zbMATH DE number 2136811
scientific article