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**1 - 3**of**3**### The Sage Project: Unifying Free Mathematical Software to Create a Viable Alternative to Magma, Maple, Mathematica and MATLAB

"... Abstract. Sage is a free, open source, self-contained distribution of mathematical software, including a large library that provides a unified interface to the components of this distribution. This library also builds on the components of Sage to implement novel algorithms covering a broad range of ..."

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Abstract. Sage is a free, open source, self-contained distribution of mathematical software, including a large library that provides a unified interface to the components of this distribution. This library also builds on the components of Sage to implement novel algorithms covering a broad range of mathematical functionality from algebraic combinatorics to number theory and arithmetic geometry.

### NEW ZERO-FREE REGIONS FOR THE DERIVATIVES OF THE RIEMANN ZETA FUNCTION

"... Abstract. The main aim of this paper is twofold. First we generalize, in a novel way, most of the known non-vanishing results for ζ (k) (s) by establishing the existence of an infinite sequence of regions in the right half-plane where these derivatives cannot have any zeros; and then, in the rare re ..."

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Abstract. The main aim of this paper is twofold. First we generalize, in a novel way, most of the known non-vanishing results for ζ (k) (s) by establishing the existence of an infinite sequence of regions in the right half-plane where these derivatives cannot have any zeros; and then, in the rare regions of the complex plane that do contain zeros of ζ (k) (s) (named “critical strips ” in analogy with the classical case of ζ(s)), we describe a unexpected phenomenon, which – especially for the hitherto-neglected high derivatives ζ (k) (s) – implies great regularities in their zero distributions. In particular, we prove sharp estimates for the number of zeros in each of these new critical strips, and we explain how they converge, in a very precise, periodic fashion, to their central, “critical ” lines, as k increases. This not only, but shows that the zeros of ζ (k)(s) are not randomly scattered to the right of the line σ = 1 2 that, in many respects, their two-dimensional distribution eventually becomes much simpler and more predictable than the one-dimensional behavior of the zeros of ζ(s) on the line σ = 1 2. 1.