nLab Euler product

Idea

The original Euler product is an expression of the Riemann zeta function as an infinite product (instead of an infinite sum as originally defined) over all prime numbers. More generally one considers Euler product expressions for general zeta functions and L-functions (various types of L-functions are defined by an Euler product, such as the Artin L-function).

When the Riemann zeta function is understood as an adelic integral via Iwasawa-Tate theory, then the existence of Euler product expressions follows from the fact that the integration measure on the idele group $\mathbb{I}_{\mathbb{Q}}$ used in this context is a product of measures on all p-adic number-components:

if a function $f\colon \mathbb{I} \longrightarrow \mathbb{R}$ has product form $f = \underset{v}{\prod} f_v$ then

$$\int_{\mathbb{I}_{\mathbb{Q}}} \; f(x) \; {\vert x\vert}^s \; d^\times x \;=\; \underset{v}{\prod} \left( \int_{\mathbb{Q}_v^\times} \;f_v(x)\; {\vert x\vert}_v^s \; d_v^\times x \right) \,.$$

(reviewed e.g. in Garrett 11, 1.6)

References

• Wikipedia, Euler product

Discussion from the point of view of adelic integration in Iwasawa-Tate theory is for instance in

• Paul Garrett, section 1.6 Iwasawa-Tate on ζ-functions and L-functions

  (pdf)