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Euler products
From OeisWiki
The infinite product expansion of a zeta function (and also of a Dirichlet L-function), indexed by the prime numbers, is called the Euler product form of that function.
Euler’s product
Euler’s product is the original Euler product, the Euler product for Euler’s zeta function (whose analytic continuation to the whole complex plane, except for a pole of order 1 ats = 1 |
-
ζ (s) := ∑ ∞n = 1
=1 n s ∏ p
p prime
=1 1 − 1 p s
⋅1 1 − 1 2 s
⋅1 1 − 1 3 s
⋅1 1 − 1 5 s
⋅1 1 − 1 7 s
⋅ ⋯, ℜ (s) > 1,1 1 − 1 11 s
p |
s = ℜ (s) + i ℑ (s) = σ + i t |
Euler product for a Dirichlet L-function
The Euler product for a Dirichlet L-function is
-
L (s, χ) := ∑ ∞n = 1
=χ (n) n s ∏ p
p prime
=1 1 − χ (p) p s
⋅1 1 − χ (2) 2 s
⋅1 1 − χ (3) 3 s
⋅1 1 − χ (5) 5 s
⋅1 1 − χ (7) 7 s
⋅ ⋯, ℜ (s) > 1,1 1 − χ (11) 11 s
p |
χ (n) |
s = ℜ (s) + i ℑ (s) = σ + i t |
χ (n) |
n |
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