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 $n \le 10^{100}$.
Generalizations of the Euler-Mascheroni constant associated with the hyperharmonic numbers
In this paper, we present two new generalizations of the Euler-Mascheroni constant arising from the Dirichlet series associated to the hyperharmonic numbers. We also give some inequalities related to upper and lower estimates, and evaluation formulas.
The first ground cricket (Orthoptera: Trigonidiidae: Nemobiinae) from mid-Cretaceous Burmese amber
Abstract A new genus and species, Birmaninemobius hirsutus gen. et sp. nov., is described on the basis of a well-preserved orthopteran specimen in mid-Cretaceous Burmese amber and attributed to the subfamily Nemobiinae (ground crickets) of the family Trigonidiidae (Grylloidea). As the first record of Nemobiinae from mid-Cretaceous Burmese amber, this new taxon not only provides novel morphological diversity in ground crickets but also indicates that the ground crickets lived in Burma (Tethyan islands) as early as the mid-Cretaceous.