A NOTE ON ζ ′ ′ ( s ) AND ζ ′ ′ ′ ( s )

There is only one pair of non-real zeros of ζ′′(s), and of ζ′′′(s), in the left half-plane. The Riemann Hypothesis implies that ζ′′(s) and ζ′′′(s) have no zeros in the strip 0 ≤ <s < 1 2 . It was shown by Speiser [3] that the Riemann Hypothesis is equivalent to ζ′(s) having no zeros in 0 < σ < 12 (as usual we write s = σ + it). Levinson and Montgomery [2] gave a different proof; moreover, they showed that ζ(s) has at most a finite number of non-real zeros in σ < 12 for k ≥ 1 as a consequence of RH. Another result in [2] was that ζ′ vanishes exactly once in the interval (−2n−2,−2n) for all n ≥ 1, these being the only zeros of ζ′ in the left half-plane. Spira [4] calculated the zeros of ζ′ and ζ′′ in the rectangle −1 ≤ σ ≤ 5, |t| ≤ 100, from which it is seen that ζ′′(s) 6= 0 in 0 ≤ σ ≤ 12 , |t| ≤ 100. However, ζ′′ has a zero near −0.36± 3.59i (which will be called b0 and b0 below). In this paper, we shall be concerned with the zeros of ζ′′(s) and ζ′′′(s) lying to the left of the critical line. Only for Theorem 1 will full details of the proof be given here (cf. [6] for Theorems 2, 3 and 4). Theorem 1. The Riemann Hypothesis implies that ζ′′(s) has no zeros in the strip 0 ≤ σ < 12 . Proof. Let us denote the real zeros of ζ′ as −an, n ≥ 1, where an ∈ (2n, 2n + 2). A non-real zero of ζ′ will be represented as ρ1 = β1 + iγ1. By what was recounted above, β1 ≥ 12 for all ρ1 (on RH), while Titchmarsh [5, Theorem 11.5(c)] showed that β1 < 3. (Since < ζ ′ ζ (s) < 0 on σ = 1 2 except when ζ(s) = 0, one has β1 = 1 2 only at a possible multiple zero of ζ(s); see [2].) The starting point is the partial fraction representation ζ′′ ζ′ (s) = ζ′′ ζ′ (0)− 2− 2 s− 1 + ∑ n ( 1 s+ an − 1 an ) + ∑ ρ1 ( 1 s− ρ1 + 1 ρ1 ) , (1) which follows from the Hadamard theory. Taking real parts in (1), we have < ′′ ζ′ (s) = ζ′′ ζ′ (0)− 2 + 2(1− σ) |s− 1|2 + ∑ n ( σ + an |s+ an| − 1 an ) + ∑ ρ1 < 1 s− ρ1 + ∑ ρ1 1 ρ1 , (2) Received by the editors November 30, 1994. 1991 Mathematics Subject Classification. Primary 11M26. c ©1996 American Mathematical Society 2311 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use