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

Abstract

Abstract. We present the zeros of the derivatives, ζ (k) (σ + it), of the Riemann zeta function for k ≤ 28 with −10 < σ < 1 2 and −10 < t < 10. Our computations show an interesting behavior of the zeros of ζ (k) , namely they seem to lie on curves which are extensions of certain chains of zeros of ζ (k) that were observed on the right half plane.

zero-free region for the fractional derivatives of the Riemann zeta function

“…The Figure 1 above gives the list of closest zeros to the pole at s = 1, and the Figure 2 depicts the distribution (and the flow) of these zeros in the left half-plane. Here, as it was noted in [5], the same phenomenon of translation of zeros continues; however, the linear and periodic movement found in the right half-plane is deformed into curves that terminate in the "trivial" zeros of derivatives of ζ(s) found on the negative real axis. The structure of the remainder of the paper is as follows.…”

“…The expression (3) for the fractional dertivatives of the Riemann zeta function will be the starting point of our proof of their zero-free regions. In order to establish the non-vanishing result, bounds on Stieltjes constants will be needed, plus a careful estimation of the behavior of the periodic Bernoulli polynomials P k (x), defined in (5). This is done in the next section.…”