MATHEMATICS OF COMPUTATION Volume 77, Number 263, July 2008, Pages 1713–1723 S 0025-5718(08)02065-6 Article electronically published on January 24, 2008 COMPUTATIONAL ESTIMATION OF THE CONSTANT β(1) CHARACTERIZING THE ORDER OF ζ (1 + it) TADEJ KOTNIK 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 →∞ ζ (2) 2β(1)e γ [1 + o(1)] log log t ≤|ζ (1 + it)|≤ 2β(1)e γ [1 + o(1)] log log t with β(1) ≥ 1 2 , while the truth of the Riemann hypothesis would also imply that β(1) ≤ 1. In the range 1 <t ≤ 10 16 , 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 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. 1. Introduction Denoting by ζ (σ + it) the Riemann zeta function, its restriction to the vertical line σ = 1 has a simple pole at t = 0 and no zeros. Away from this pole, |ζ (1 + it)| is even and continuous, and both |ζ (1 + it)| and 1/ |ζ (1 + it)| are unbounded, so that as t increases, |ζ (1 + it)| takes arbitrarily large values, as well as values arbitrarily close to zero. An illustration of this behavior is shown in Figure 1, and its more precise formulation is based on two inequalities due to Norman Levinson. Improving upon previous work by Bohr and Landau [1], Littlewood [2], [3], Titchmarsh [4], [5], and Chowla [6], Levinson showed in 1972 [7] that each of the two inequalities (1) |ζ (1 + it)|≤ ζ(2) e γ (log log t−log log log t) and |ζ (1 + it)|≥ e γ log log t holds unconditionally for an infinite number of arbitrarily large values of t. In an arXiv preprint published in 2005, Granville and Soundararajan [8] report that in the denominator of the first of these inequalities the term log log log t can be im- proved to O(1), while in the second inequality the term log log t can be improved to log log t + log log log t − log log log log t + O(1). As will be discussed later, this second improvement is essential for the interpretation of the numerical data acquired in this paper. Received by the editor August 15, 2006 and, in revised form, April 26, 2007. 2000 Mathematics Subject Classification. Primary 11M06, 11Y60; Secondary 11Y35, 65A05. Key words and phrases. Riemann’s zeta function, line σ = 1, constant β(1). c 2008 American Mathematical Society Reverts to public domain 28 years from publication 1713