Euler Product Formula

Arithmetic Structure of the Zeta Function

The defining series of the Riemann zeta function is

\zeta(s) = \sum_{n=1}^{\infty}\frac1{n^s}, \qquad \operatorname{Re}(s)>1.

At first sight, this appears to be merely an analytic object: an infinite series over the positive integers. Its true arithmetic significance emerges through the Euler product formula, which expresses the same function as a product over primes.

This identity is one of the foundational results of analytic number theory because it links prime factorization with analytic behavior.

Geometric Series at a Prime

Fix a prime number $p$ . For

\operatorname{Re}(s)>1,

the geometric series converges:

1+\frac1{p^s}+\frac1{p^{2s}}+\frac1{p^{3s}}+\cdots = \left(1-\frac1{p^s}\right)^{-1}.

Thus every prime contributes a local factor

\left(1-\frac1{p^s}\right)^{-1}.

Multiplying these factors over all primes gives the formal product

\prod_p \left(1-\frac1{p^s}\right)^{-1}.

The remarkable fact is that this product equals the zeta series itself.

Statement of Euler’s Formula

For

\operatorname{Re}(s)>1,

the Riemann zeta function satisfies

\zeta(s) = \prod_p \left(1-\frac1{p^s}\right)^{-1}.

The product extends over all prime numbers.

This identity is called Euler’s product formula.

Proof of the Formula

Expand the finite product

\prod_{p\leq y} \left( 1+\frac1{p^s}+\frac1{p^{2s}}+\cdots \right).

To form a term in the expansion, choose one power of each prime. A typical term has the form

\frac1 { p_1^{a_1s} p_2^{a_2s} \cdots p_k^{a_ks} }.

By unique factorization,

n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}

represents exactly one positive integer $n$ . Therefore the expanded product equals

\sum_{\substack{n\geq1\\ p\mid n\Rightarrow p\leq y}} \frac1{n^s}.

As $y\to\infty$ , every positive integer eventually appears. Since the series converges absolutely for

\operatorname{Re}(s)>1,

the limit gives

\zeta(s) = \prod_p \left(1-\frac1{p^s}\right)^{-1}.

The proof depends entirely on the Fundamental Theorem of Arithmetic.

Prime Numbers and Divergence

Euler used the product formula to give a new proof that infinitely many primes exist.

Suppose only finitely many primes existed. Then the product

\prod_p \left(1-\frac1p\right)^{-1}

would contain finitely many finite factors and therefore converge.

However,

\zeta(1) = \sum_{n=1}^{\infty}\frac1n

diverges. Thus infinitely many primes must exist.

This argument was one of the first major applications of analytic methods to arithmetic.

Logarithmic Form

Taking logarithms of the Euler product gives

\log\zeta(s) = -\sum_p \log\left(1-\frac1{p^s}\right).

Using the expansion

-\log(1-x) = x+\frac{x^2}{2}+\frac{x^3}{3}+\cdots,

one obtains

\log\zeta(s) = \sum_p \sum_{k=1}^{\infty} \frac1{k\,p^{ks}}.

The dominant contribution comes from the first powers:

\sum_p \frac1{p^s}.

This relation connects the analytic behavior of $\zeta(s)$ directly with prime sums.

Reciprocal of the Zeta Function

The Euler product immediately yields

\frac1{\zeta(s)} = \prod_p \left(1-\frac1{p^s}\right).

Expanding the product gives

\frac1{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s},

where $\mu(n)$ is the Möbius function.

Thus the reciprocal of the zeta function encodes Möbius inversion and multiplicative cancellation.

Local-to-Global Principle

Euler products express a central philosophy of number theory:

global arithmetic information decomposes into local prime data.

Each prime contributes an individual factor

\left(1-p^{-s}\right)^{-1}.

The full arithmetic structure emerges by combining all primes multiplicatively.

This principle extends far beyond the zeta function. Dirichlet $L$ -functions, Dedekind zeta functions, modular $L$ -functions, and automorphic $L$ -functions all possess Euler products.

Convergence of the Product

The Euler product converges absolutely for

\operatorname{Re}(s)>1.

Indeed,

\sum_p \frac1{|p^s|} = \sum_p \frac1{p^\sigma}

converges when

\sigma>1.

Absolute convergence justifies rearrangements and analytic manipulations.

Near the line

\operatorname{Re}(s)=1,

the product becomes much more delicate. The divergence of

\sum_p \frac1p

reflects the singularity of the zeta function at $s=1$ .

Importance

The Euler product formula is the bridge between analysis and arithmetic.

The series definition of $\zeta(s)$ involves all positive integers. The Euler product isolates the prime numbers hidden inside that series. Because of this identity, analytic properties of the zeta function translate into statements about primes.

Much of analytic number theory consists of studying this translation:

poles correspond to prime density,
zeros correspond to prime fluctuations,
logarithmic derivatives produce weighted prime sums,
analytic continuation reveals hidden arithmetic structure.

The Euler product therefore forms the conceptual foundation of the analytic theory of prime numbers.