Dirichlet Eta Function

$$\eta(x) \equiv \sum_{n=1}^\infty (-1)^{n-1}n^{-x} = (1-2^{1-x})\zeta(x),$$ (1)

where $n=1$, 2, ..., and $\zeta(x)$ is the Riemann Zeta Function. Particular values are given in Abramowitz and Stegun (1972, p. 811). The eta function is related to the Riemann Zeta Function and Dirichlet Lambda Function by

$${\zeta(\nu)\over 2^\nu}={\lambda(\nu)\over 2^\nu-1}={\eta(\nu)\over 2^{\nu}-2}$$ (2)

and

$$\zeta(\nu)+\eta(\nu)=2\lambda(\nu)$$ (3)

(Spanier and Oldham 1987). The value $\eta(1)$ may be computed by noting that the Maclaurin Series for $\ln(1+x)$ for $-1<x\leq 1$ is

$$\ln(1+x)=x-{\textstyle{1\over 2}}x^2+{\textstyle{1\over 3}}x^3-{\textstyle{1\over 4}}x^4+\ldots.$$ (4)

Therefore,

$$\ln 2 = \ln(1+1) = 1-{\textstyle{1\over 2}}+{\textstyle{1\over 3}}-{\textstyle{1\over 4}}+\ldots$$

$$= \sum_{n=1}^\infty {(-1)^{n-1}\over n} = \eta(1).$$ (5)

Values for Even Integers are related to the analytical values of the Riemann Zeta Function. $\eta(0)$ is defined to be ${\textstyle{1\over 2}}$.

$$\eta(0) = {\textstyle{1\over 2}}$$

$$\eta(1) = \ln 2$$

$$\eta(2) = {\pi^2\over 12}$$

$$\eta(3) = 0.90154\dots$$

$$\eta(4) = {7\pi^4\over 720}.$$

See also Dedekind Eta Function, Dirichlet Beta Function, Dirichlet Lambda Function, Riemann Zeta Function, Zeta Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 807-808, 1972.

Spanier, J. and Oldham, K. B. ``The Zeta Numbers and Related Functions.'' Ch. 3 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 25-33, 1987.