Another Zero-Free Region for ζ (k) (s)

More Zeros of the Derivatives of the Riemann Zeta Function on the Left Half Plane

“…In this paper we numerically investigate the distribution of zeros of the derivatives ζ (k) of ζ on the left half plane. The results of our computations, that considerably expands the list of previously published zeros [11,15], can be found in Table 1 and Table 2. For the rectangular region −10 < σ < 1 2 and |t| < 10, Table 1 contains the number of zeros of ζ (k) , its real zeros, and its zeros with 0 < σ < 1 2 .…”

“…For k ≥ 3 such general upper bounds were given by Spira [11] and later improved by Verma and Kaur [14]: where q 2 is given by the formula…”

“…Spira [11] computed zeros of the first and second derivative of ζ(s) for 0 < t < 100 and noticed that they occur in pairs. Skorokhodov [9] went further in his computation and noticed that the zeros of derivatives of ζ seem to form chains, that is for each zero z (k) of ζ (k) there seems to be a corresponding zero z (k+1) of ζ (k+1) .…”

On vertical zeros of Re ζ(s) and Im ζ(s)

“…where n 1 , n 2 are integers, using (4) and applying an argument of R. Spira [4], we can prove that F (s) and F (s) − ζ(σ 0 + s) have the same number of zeros in the rectangle with vertices at M ± iT 0 , T ± iT 0 . Indeed, on the sides of this rectangle due to the Γ -factor we have Remark.…”

Formulas for Higher Derivatives of the Riemann Zeta Function

“…where the coefficients ajkm and bjkm are independent of s. This formula was used by Spira [11], [12] to determine zero-free regions for Ç(k)(s), and by Berndt [5], to determine the asymptotic number of zeros of £(fc)(s) with 0 < t < T, where s = a + it. This paper gives a variant of this formula (Theorem 1) which enables us to determine the coefficients ajkm and bjkm explicitly (Theorem 2).…”