Euler Products and Multiplicativity

Euler products reveal the profound connection between additive structure (summing over integers) and multiplicative structure (products over primes), forming the heart of analytic number theory.

DefinitionEuler Product

A Dirichlet series $F(s) = \sum_{n=1}^{\infty} a_n/n^s$ has an Euler product if it can be written as:

F(s) = \prod_{p \text{ prime}} F_p(s)

where $F_p(s)$ depends only on the prime $p$ . This holds when $a_n$ is multiplicative.

TheoremEuler Product for Multiplicative Functions

If $f$ is a multiplicative arithmetic function and $F(s) = \sum_{n=1}^{\infty} f(n)/n^s$ converges absolutely for $\Re(s) > \sigma$ , then:

F(s) = \prod_{p} \left(1 + \frac{f(p)}{p^s} + \frac{f(p^2)}{p^{2s}} + \frac{f(p^3)}{p^{3s}} + \cdots \right)

If $f$ is completely multiplicative:

F(s) = \prod_{p} \frac{1}{1 - f(p)p^{-s}}

ExampleClassical Euler Products

Zeta function: $\zeta(s) = \prod_{p} \frac{1}{1 - p^{-s}}$ (for $\Re(s) > 1$ )
Reciprocal zeta: $\frac{1}{\zeta(s)} = \prod_{p} (1 - p^{-s}) = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s}$
Totient: $\frac{\zeta(s-1)}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\varphi(n)}{n^s} = \prod_{p} \frac{1 - p^{-s}}{1 - p^{1-s}}$

The Euler product for $\zeta(s)$ immediately proves the infinitude of primes: if there were finitely many primes, the right side would be a finite product, but the left side has a pole at $s=1$ .

DefinitionLogarithmic Derivative

Taking the logarithmic derivative of an Euler product reveals prime information:

-\frac{\zeta'(s)}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s}

where $\Lambda(n)$ is the von Mangoldt function:

\Lambda(n) = \begin{cases} \log p & \text{if } n = p^k \text{ for prime } p \\ 0 & \text{otherwise} \end{cases}

ExamplePrime Number Theorem Connection

The Prime Number Theorem ( $\pi(x) \sim x/\log x$ ) is equivalent to:

\sum_{p \leq x} 1 \sim \frac{x}{\log x}

Using partial summation and the Wiener-Ikehara theorem, this reduces to showing:

\frac{\zeta'(s)}{\zeta(s)} = -\sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s}

has a simple pole at $s=1$ and extends holomorphically to $\Re(s) = 1$ .

Remark

Euler products encode the Fundamental Theorem of Arithmetic (unique prime factorization) in analytic form. Each prime contributes independently, and the product structure mirrors the multiplicative nature of integers.

The Euler product philosophy—that global properties of integers reflect local properties at each prime—pervades modern number theory, from L-functions to the Langlands program.