Euler's alternating zeta function

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Euler defined the alternating zeta function

ϕ (s)

, also denoted

ζ  * (s)

, of a real variable

s

. The alternating zeta function is also known as the Dirichlet eta function

η (s)

. It is defined by the following alternating series (which converges for

s > 0

)

ϕ (s) :=   ∞ ∑   n   = 1    (−1) n +1 n  s = 1 1 s − 1 2 s + 1 3 s − 1 4 s + ⋯, s > 0.

Euler wanted to calculate

ϕ (s)

for

s =  −  m

with

m = 0, 1, 2, 3, ...

and to do so he introduced the Eulerian polynomials. This series converges over the complex plane for

ℜ(s) > 0

. The function

ϕ (s)

can be analytically continued (holomorphically extended) to the whole complex plane.

Functional equation

The alternating zeta function is related to the zeta function

ζ (s)

by the functional equation

ϕ (s)  =  2 s  − 1 − 1 2 s  − 1 ζ (s)  =  (1 − 2 1 − s ) ζ (s), s ≠ 1.

Some values of

ϕ (s)

ϕ (0) = (1 − 2 1 −  0  ) ζ  (0) = − (− 1 2 ) = 1 2 . ϕ (1) =   ∞ ∑   n   = 1    (−1) n +1 n = log (2), from log (1 + x) =   ∞ ∑   n   = 1    (−1) n +1 n x n, −1 < x ≤ 1, with x = 1. ϕ (2) = 1 2 ζ  (2) = 1 2  ⋅   π 2 6 = 1 12 π 2. ϕ (3) = 3 4 ζ  (3). ϕ (4) = 7 8 ζ  (4) = 7 8  ⋅   π 4 90 = 7 720 π 4. ϕ (5) = 15 16 ζ (5).

See also

• Euler’s zeta function
• Riemann zeta function
• Euler products
• Euler’s alternating zeta function
• Prime zeta function
• Hurwitz zeta function
• Multivariate zeta function

• Dirichlet L-functions