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 $n \le 10^{100}$.
Erratum and Addendum to: Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results [Ramanujan J. (2014), 35:21–110]
The historical analysis of functional Eqs. (20)–(22) on pp. 35–37 is far fromexhaustive. In order to give a larger vision of this subject, several complimentary remarks may be needed. First, on p. 37, lines 1–5, the text “By the way, the above reflection formula (21) for L(s)was also obtained by Oscar Schlömilch; in 1849 he presented it as an exercise for students [55], and then, in 1858, he published the proof [56]. Yet, it should be recalled that an analog of formula (20) for the alternating...”may be replaced by the following one: “By the way, between 1849 and 1858, the above reflection formula for L(s) was also obtained by several other mathematicians, including Oscar Schlömilch [55, 56], Gotthold Eisenstein [73, 84], and Thomas Clausen [75].1 Yet, it should be noted that formula (20) itself was rigorously proved by Kinkelin a year before Riemann [79, p. 100], [78], and its analog for the alternating...”
On the summation of some divergent series of Euler and the zeta functions
Grâce aux travaux d'Euler (L.) traitant des techniques analytiques et algorithmiques, la somme des series divergentes peut, d'une part, etre evaluee numeriquement puisqu'aucune solution n'est apparamment connue, puis analytiquement