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The behaviour of the Riemann zeta-function on the critical line

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  26 February 2010

R. R. Hall
R. R. Hall
Affiliation:
Department of Mathematics, University of York, York, YO1 5DD.
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Extract

We are interested in the distribution of those zeros of the Riemann zeta-function which lie on the critical line ℜs = ½, and the maxima of the function between successive zeros. Our results are to be independent of any unproved hypothesis. Put

MSC classification

Secondary: 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses
Type
Research Article
Information
Mathematika , Volume 46 , Issue 2 , December 1999 , pp. 281 - 313
Copyright
Copyright © University College London 1999

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References

1. Beesack, P. R.. Hardy's inequality and its extensions. Pacific J. Math. 11 (1961), 39–61.CrossRefGoogle Scholar
2. Conrey, J. B.. The fourth moment of derivatives of the Riemann zeta-function. Quart. J. Oxford, 39 (1988), 21–36.CrossRefGoogle Scholar
3. Conrey, J. B. and Ghosh, A.. A mean value theorem for the Riemann zeta-function at its relative extrema on the critical line. J. London Math. Soc: (2), 32 (1985), 193–202.CrossRefGoogle Scholar
4. Conrey, J. B., Ghosh, A. and Gonek, S. M.. A note on gaps between zeros of the zeta-function. Bull. London Math. Soc, 16 (1984), 421–424.CrossRefGoogle Scholar
5. Conrey, J. B., Ghosh, A. and Gonek, S. M.. Large gaps between zeros of the zeta-function. Mathematika, 33 (1986), 212–238.CrossRefGoogle Scholar
6. Conrey, J. B., Ghosh, A., Gonek, S. M. and Heath-Brown, D. R.. On the distribution of gaps between zeros of the zeta-function. Quart. J. Math. Oxford (2), 36 (1985), 43–51.CrossRefGoogle Scholar
7. Gonek, S. M.. Mean values of the Riemann zeta-function and its derivatives. Invent, math. 75 (1984), 123–141.CrossRefGoogle Scholar
8. Hardy, G. H. and Littlewood, J. E.. Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes. Ada Math., 41 (1918), 119–196.Google Scholar
9. Hardy, G. H. and Littlewood, J. E.. The approximate functional equations for ζ(s) and ζ2(s). Proc. London Math. Soc. (2), 29 (1929), 81–97.CrossRefGoogle Scholar
10. Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities. (Cambridge, 1934).Google Scholar
11. Heath-Brown, D. R.. The fourth power moment of the Riemann zeta-function. Proc. London Math. Soc. (3), 38 (1979), 385–422.CrossRefGoogle Scholar
12. Ingham, A. E.. Mean-value theorems in the theory of the Riemann zeta-function. Proc. London Math. Soc. (2), 27 (1928), 273–300.CrossRefGoogle Scholar
13. Ivic, A.. The Riemann zeta-function. (New York, 1985).Google Scholar
14. Selberg, A.. The zeta-function and the Riemann hypothesis. Skandinaviske Mathcmatikcrkongres, 10 (1946), 187–200.Google Scholar
15. Titchmarsh, E. C.. The theory of the Riemann zeta-function (Revised by D. R. Heath-Brown). (Oxford, 1986).Google Scholar
16. Titchmarsh, E. C.. The approximate functional equation for ζ2(s). Quart. J. Oxford, 9 (1938), 109–114.CrossRefGoogle Scholar
17. Zavorotnyi, N. I.. On the fourth moment of the Riemann zeta-function (in Russian). In Automorphic Forms and Number Theory 1 (Coll. Sci. Vladivostok, 1989), pp. 69–185.Google Scholar