Euler Product

An Euler product is an infinite product expansion of a Dirichlet series over the prime numbers, first introduced by Leonhard Euler in his proof of the divergence of the sum of prime reciprocals. For the Riemann zeta function, the Euler product ζ(s) = ∏_p (1 − p^{−s})^{−1} encodes the fundamental theorem of arithmetic in analytic form: the unique factorization of integers into primes becomes the factorization of a global L-function into local Euler factors. Every Dedekind zeta function and every Dirichlet L-function admits such a product, and the Langlands program conjectures that all automorphic L-functions do as well.

The Euler product is not a computational convenience. It is the analytic manifestation of the local-global principle: each prime contributes one factor, and the global object is the product of all local contributions. To study an L-function without its Euler product is to study a symphony by reading the score one note at a time.