Some bounds and limits in the theory of Riemann's zeta function
For any real a > 0 we determine the supremum of the real σ such that ζ(σ+it) = a for some real t. For 0 < a < 1, a = 1, and a > 1 the results turn out to be quite different. We also determine the supremum E of the real parts of the ‘turning points’, that is points σ + it where a curve Im ζ(σ + it) =...
| Authors: | ; |
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| Format: | Article |
| Status: | Submitted version |
| Publication date: | 2012 |
| Country: | España |
| Institution: | Universidad de Sevilla (US) |
| Repository: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/42131 |
| Online Access: | http://hdl.handle.net/11441/42131 https://doi.org/10.1016/j.jmaa.2012.06.017 |
| Access Level: | Open access |
| Keyword: | zeta function LLL algorithm extreme values |
| Summary: | For any real a > 0 we determine the supremum of the real σ such that ζ(σ+it) = a for some real t. For 0 < a < 1, a = 1, and a > 1 the results turn out to be quite different. We also determine the supremum E of the real parts of the ‘turning points’, that is points σ + it where a curve Im ζ(σ + it) = 0 has a vertical tangent. This supremum E (also considered by Titchmarsh) coincides with the supremum of the real σ such that ζ 0 (σ + it) = 0 for some real t. We find a surprising connection between the three indicated problems: ζ(s) = 1, ζ 0 (s) = 0 and turning points of ζ(s). The almost extremal values for these three problems appear to be located at approximately the same height. |
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