Zero-Free Regions of ζ(^k )(s)

On the stationary points of Hardy's function Z(t)

“…There are a number of results in the literature concerning the zeros of ζ (k) (s) and ξ (k) (s): we mention Speiser [10], Berndt [1], Spira [11]- [14], Levinson [8] and Levinson and Montgomery [9]. As far as we are aware, Z (w) has not been studied directly before now, however it appears in a slightly disguised form in Conrey and Ghosh [3] where these authors prove the following formula (conditional on the Riemann Hypothesis):…”

On the a-points of derivatives of the Riemann zeta function

“…When a = 0, Titchmarsh [12, Theorem 11.5(C)] and Spira [9] have already proved this theorem. Hence we only prove the case a = 0.…”

“…When a = 0, Spira [9] found the zero free region for ζ (k) (s). The next theorem is a generalization of his result.…”

A note on 𝜁”(𝑠) and 𝜁”’(𝑠)

“…The intervals where these zeros lie can be found from the functional equation of ζ(s) differentiated twice. From Spira [4] we know that β 2 < 5 for all ρ 2 , and analogous to (4), by using (5) at s = 10, we find If one determines the 'small' zeros of ζ (s), including the pair in the left halfplane, then one may proceed to the investigation of ζ (iv) (s).…”

“…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) = 0 in 0 ≤ σ ≤ 1 2 , |t| ≤ 100. However, ζ has a zero near −0.36 ± 3.59i (which will be called b 0 and b 0 below).…”