A Wirtinger type inequality and the spacing of the zeros of the Riemann zeta-function. (English) Zbl 0994.11030
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 J. Conrey, A. Ghosh and 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 \(\nu\geq 0\).
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 \(\nu\geq 0\).
Reviewer: Roger Heath-Brown (Oxford)
MSC:
| 11M26 | Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses |
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
Citations:
Zbl 0536.10033References:
| [1] | Beesack, P. R., Hardy’s inequality and its extensions, Pacific J. Math., 11, 39-61 (1961) · Zbl 0103.03503 |
| [2] | Conrey, J. B., The fourth moment of derivatives of the Riemann zeta-function, Quart. J. Math. Oxford Ser. (2), 39, 21-36 (1988) · Zbl 0644.10028 |
| [3] | J. B. Conrey, and, A. Ghosh, On mean values of the zeta-function, III, in, Proceedings of the Amalfi conference in Analytic Number Theory, Universita di Salerno, 1992.; J. B. Conrey, and, A. Ghosh, On mean values of the zeta-function, III, in, Proceedings of the Amalfi conference in Analytic Number Theory, Universita di Salerno, 1992. · Zbl 0792.11033 |
| [4] | Conrey, J. B.; Ghosh, A.; Gonek, S. M., A note on gaps between zeros of the zeta-function, Bull. London Math. Soc., 16, 421-424 (1984) · Zbl 0536.10033 |
| [5] | Conrey, J. B.; Ghosh, A.; Gonek, S. M., Large gaps between zeros of the zeta-function, Mathematika, 33, 212-238 (1986) · Zbl 0615.10048 |
| [6] | J. B. Conrey, and, S. M. Gonek, preprint.; J. B. Conrey, and, S. M. Gonek, preprint. |
| [7] | Hall, R. R., The behavior of the Riemann zeta-function on the critical line, Mathematika, 46, 281-313 (1999) · Zbl 1134.11337 |
| [8] | Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities (1934) · JFM 60.0169.01 |
| [9] | Ingham, A. E., Mean-value theorems in the theorem of the Riemann zeta-function, Proc. London Math. Soc. (2), 27, 273-300 (1928) · JFM 53.0313.01 |
| [10] | Ivić, A., The Riemann Zeta-Function (1985) · Zbl 0583.10021 |
| [11] | Keating, J. P.; Snaith, N. C., Random matrix theory and \(ζ(12+ it )\), Comm. Math. Phys., 214, 57-89 (2000) · Zbl 1051.11048 |
| [12] | Montgomery, H. L.; Odlyzko, A. M., Gaps between zeros of the zeta-function, Colloq. Math. Soc. János Bolyai (1981), North-Holland: North-Holland Amsterdam · Zbl 0546.10033 |
| [13] | Selberg, A., The zeta-function and the Riemann Hypothesis, Skand. Math., 10, 187-200 (1946) · Zbl 0030.05003 |
| [14] | Whittaker, E. T.; Watson, G. N., Modern Analysis (1958) · Zbl 0108.26903 |
| [15] | Zavorotnyi, N. I., On the fourth moment of the Riemann zeta-function, Automorphic Forms and Number Theory 1 (1989), Coll. Sci: Coll. Sci Vladivostok, p. 69-185 · Zbl 0711.11028 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
