Euler products

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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 at

s = 1

, is known as the Riemann zeta function)

ζ (s) := ∑ ∞ n   = 1 1 n s =   p p prime ∏   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  ⋅   1 1 − 1 11 s  ⋅   ⋯, ℜ (s) > 1,

the product being over the primes

p

and

s = ℜ (s) + i ℑ (s) = σ + i t

(this notation originating from Bernhard Riemann).

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 ∏   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  ⋅   1 1 − χ (11) 11 s  ⋅   ⋯, ℜ (s) > 1,

the product being over the primes

p

, where

χ (n)

is a Dirichlet character and

s = ℜ (s) + i ℑ (s) = σ + i t

(this notation originating from Bernhard Riemann). The Riemann zeta function is the special case obtained when

χ (n)

is 1 for all

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

• Dirichlet L-functions