alternating zeta function | eta function

The Dirichlet eta function is the function η(s) defined by η(s) | congruent | sum_(k = 1)^∞ (-1)^(k - 1)/k^s | = | (1 - 2^(1 - s)) ζ(s), where ζ(s) is the Riemann zeta function. Note that Borwein and Borwein use the notation α(s) instead of η(s). The function is also known as the alternating zeta function and denoted ζ^*(s). η(0) = 1/2 is defined by setting s = 0 in the right-hand side of (-1), while η(1) = ln2 (sometimes called the alternating harmonic series) is defined using the left-hand side. The function vanishes at each zero of 1 - 2^(1 - s) except s = 1.

Dedekind eta function | Dirichlet beta function | Dirichlet lambda function | Hadjicostas's formula | Riemann zeta function | zeta function