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Computing Stieltjes constants using complex integration
Computing Stieltjes constants using complex integration
- Source :
- Mathematics of Computation, Mathematics of Computation, American Mathematical Society, 2019, 88 (318), ⟨10.1090/mcom/3401⟩, Mathematics of Computation, 2019, 88 (318), ⟨10.1090/mcom/3401⟩
- Publication Year :
- 2018
- Publisher :
- HAL CCSD, 2018.
-
Abstract
- International audience; The generalized Stieltjes constants $\gamma_n(v)$ are, up to a simple scaling factor, the Laurent series coefficients of the Hurwitz zeta function $\zeta(s,v)$ about its unique pole $s = 1$. In this work, we devise an efficient algorithm to compute these constants to arbitrary precision with rigorous error bounds, for the first time achieving this with low complexity with respect to the order~$n$. Our computations are based on an integral representation with a hyperbolic kernel that decays exponentially fast. The algorithm consists of locating an approximate steepest descent contour and then evaluating the integral numerically in ball arithmetic using the Petras algorithm with a Taylor expansion for bounds near the saddle point. An implementation is provided in the Arb library. We can, for example, compute $\gamma_n(1)$ to 1000 digits in a minute for any $n$ up to $n=10^{100}$. We also provide other interesting integral representations for $\gamma_n(v)$, $\zeta(s)$, $\zeta(s,v)$, some polygamma functions and the Lerch transcendent.
- Subjects :
- rigorous error bounds
2010 Mathematics Subject Classification. Primary 11M35, 65D20
Secondary 65G20
arbitrary-precision arithmetic
[ MATH.MATH-CA ] Mathematics [math]/Classical Analysis and ODEs [math.CA]
[ INFO.INFO-NA ] Computer Science [cs]/Numerical Analysis [cs.NA]
Laurent series
Stieltjes constants
010103 numerical & computational mathematics
[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA]
01 natural sciences
Hurwitz zeta function
symbols.namesake
Saddle point
Arbitrary-precision arithmetic
Classical Analysis and ODEs (math.CA)
FOS: Mathematics
Taylor series
Riemann zeta function
Ball (mathematics)
0101 mathematics
Mathematics
Discrete mathematics
Algebra and Number Theory
Applied Mathematics
010102 general mathematics
Computer Science - Numerical Analysis
Numerical Analysis (math.NA)
[INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA]
integral representation
Computational Mathematics
Mathematics - Classical Analysis and ODEs
symbols
numerical integration
complexity
complex integration
Subjects
Details
- Language :
- English
- ISSN :
- 00255718
- Database :
- OpenAIRE
- Journal :
- Mathematics of Computation, Mathematics of Computation, American Mathematical Society, 2019, 88 (318), ⟨10.1090/mcom/3401⟩, Mathematics of Computation, 2019, 88 (318), ⟨10.1090/mcom/3401⟩
- Accession number :
- edsair.doi.dedup.....4b71429fcf4a679cc77e8865f1681c25
- Full Text :
- https://doi.org/10.1090/mcom/3401⟩