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On the modulus of the Riemann zeta function in the critical strip. (English) Zbl 1048.11069
Summary: For the Riemann zeta-function \(\zeta (s)\), defined for complex \(s = \sigma + \operatorname {i}\kern -1pt t\), we write \(\sigma = \frac {1}{2} + \Delta \), and we study the horizontal behaviour of \(| \zeta (s)| \) in the critical strip \(| \Delta | \leq \frac {1}{2}\). We prove \[ \biggl | \zeta \biggl (\tfrac 12-\Delta + \operatorname {i}t \biggr )\biggr | \geq \biggl | \zeta \biggl (\tfrac 12+\Delta + \operatorname {i}t \biggr )\biggr | \] for \(0 \leq \Delta \leq \frac {1}{2}\), \(2\pi + 1 \leq t\); and we give accurate but simple asymptotic estimates for the quotient \(\alpha (\Delta , t)\) of these two quantities. Inequalities and numerical tables are presented which show just how accurate these estimates are. Several conjectures related to the Riemann Hypothesis are discussed as well.

MSC:
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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