A Wirtinger type inequality and the spacing of the zeros of the Riemann zeta-function
Let \(Z(t)= e^{i\theta(t)} \zeta(1/2+it)\) be the usual \(Z\)-function, real valued for real \(t\). Then it is shown unconditionally that the successive real zeros \(t_n\) of \(Z(t)\) satisfy \[ \limsup \frac{t_{n+1}-t_n} {2\pi/\log t_n}\geq \biggl( \frac{11}{2} \biggr)^{1/2}- 2.345207\dots\;. \] In a previous paper [Mathematika 46, 281-313 (1999)], the author obtained the slightly smaller lower bound \((105/4)^{1/4}= 2.263509\dots\;\). Under the generalized Riemann hypothesis the lower bound 2.68 has been obtained by \textit{J. Conrey, A. Ghosh} and \textit{S. Gonek} [Bull. Lond. Math. Soc. 16, 421-424 (1984; Zbl 0536.10033)]. For the proof one uses asymptotic formulae for the mean values of \(Z^4,Z^2, Z^{\prime 2} Z^{\prime 4}\) and \(Z^2 Z^{\prime\prime 2}\). These are combined with a general lower bound inequality \[ \int_0^\pi \{y'(x)^4+ 6\nu y(x)^2 y'(x)^2\} dx\geq 3\lambda_0(\nu) \int_0^\pi y(x)^4 dx \] with \[ \lambda_0(\nu)= \tfrac{1}{8} \{1+4\nu+ \sqrt{1+8\nu}\}, \] which holds for any function \(y(x)\in C^2[0,\pi]\) with \(y(0)= y(\pi)=0\), and for any constant \(\nu\geq 0\).
scientific article; zbMATH DE number 1765875
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