Dirichlet eta function η(s)¶
The Dirichlet eta function is the alternating Dirichlet series
\[
\eta(s)=\sum_{n\ge 1} \frac{(-1)^{n-1}}{n^s},
\]
\[
\eta(s) = \left(1-2^{1-s}\right)\zeta(s).
\]
Key ideas¶
Faster convergence: for real (s>1), the alternating series often converges more rapidly than the plain ζ-series.
Analytic continuation: the identity above provides an analytic continuation of ζ(s) away from the factor (1-2^{1-s}).
Numerical gateway: η(s) is a convenient “entry point” for simple ζ(s) numerics without complex contour methods.
Why it matters in this project¶
When we want quick, small-scale experiments (e.g. comparing partial sums, smoothing, or convergence acceleration), η(s) provides stable numerics with minimal infrastructure.
Experiments in this repository¶
E114 — ζ(1/2+it) via η(s): truncation/precision stability map.
References¶
See References.
[Edwards, 1974, Titchmarsh, 1986, Wikipedia contributors, 2025]