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Computational Estimation of the Constant β(1) Characterizing the Order of ζ(1+it)

Tadej Kotnik
Mathematics of Computation
Vol. 77, No. 263 (Jul., 2008), pp. 1713-1723
Stable URL: http://www.jstor.org/stable/40234579
Page Count: 11
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Computational Estimation of the Constant β(1) Characterizing the Order of ζ(1+it)
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Abstract

The paper describes a computational estimation of the constant β(1) characterizing the bounds of |ζ(1 + it)|. It is known that as t → ∞ ${{\zeta (2)} \over {2\beta (1)e^\gamma [1 + o(1)]\log \log t}} \le |\zeta (1 + it)| \le 2\beta (1)e^\gamma [1 + o(1)]\log \log t$ with $\beta (1) \ge {1 \over 2},$ while the truth of the Riemann hypothesis would also imply that β(1) ≤ 1. In the range 1 < t ≤ 10¹⁶, two sets of estimates of β(1) are computed, one for increasingly small minima and another for increasingly large maxima of |ζ(1 + it)|. As t increases, the estimates in the first set rapidly fall below 1 and gradually reach values slightly below 0.70, while the estimates in the second set rapidly exceed ${1 \over 2}$ and gradually reach values slightly above 0.64. The obtained numerical results are discussed and compared to the implications of recent theoretical work of Granville and Soundararajan.

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