Computing Stieltjes constants using complex integration
The Stieltjes constants $\gamma\_n$ are the coefficients appearing in the Laurent series of the Riemann zeta function at $s = 1$. We give a simple and efficient method to compute a $p$-bit approximation of $\gamma\_n$ with rigorous error bounds. Starting from an integral representation due to Blagouchine, we shift the contour to eliminate cancellation. The integral is then evaluated numerically in ball arithmetic using the Petras algorithm, with the use of a Taylor expansion for bounds near the saddle point. This appears to be the first algorithm for Stieltjes constants with uniformly low complexity with respect to both $n$ and $p$. An implementation is provided in the Arb library. We can, for example, compute $\gamma\_n$ to 1000 digits in a minute for any $n \le 10^{100}$.
An introduction to the Bernoulli function
The Bernoulli function $\operatorname{B}(s, v) = - s\, \zeta(1-s, v)$ interpolates the Bernoulli numbers but can be introduced independently of the zeta function. The point of departure is a modification of the Stieltjes constants based on an integral representation given by J. Jansen. The functional equation of $\operatorname{B}(s, v)$ and its relation to the Riemann $\zeta$ and $\xi$ function is explored. Classical results of Hadamard, Worpitzky, and Hasse are recast in terms of $\operatorname{B}(s, v).$ The extended Bernoulli function defines the Bernoulli numbers for odd indices harmonizing with rational numbers studied by Euler in 1735 and which are the bridge to the Euler and Andre numbers. Interpolating functions for both the signed and the unsigned case are given. The Swiss knife polynomials let the integer sequences of the Euler-Bernoulli family calculate easily.
Live birth in the Devonian period
The extinct placoderm fishes were the dominant group of vertebrates throughout the Middle Palaeozoic era, yet controversy about their relationships within the gnathostomes (jawed vertebrates) is partly due to different interpretations of their reproductive biology. Here we document the oldest record of a live-bearing vertebrate in a new ptyctodontid placoderm, Materpiscis attenboroughi gen. et sp. nov., from the Late Devonian Gogo Formation of Australia (approximately 380 million years ago). The new specimen, remarkably preserved in three dimensions, contains a single, intra-uterine embryo connected by a permineralized umbilical cord. An amorphous crystalline mass near the umbilical cord possibly represents the recrystallized yolk sac. Another ptyctodont from the Gogo Formation, Austroptyctodus gardineri, also shows three small embryos inside it in the same position. Ptyctodontids have already provided the oldest definite evidence for vertebrate copulation, and the new specimens confirm that some placoderms had a remarkably advanced reproductive biology, comparable to that of some modern sharks and rays. The new discovery points to internal fertilization and viviparity in vertebrates as originating earliest within placoderms.