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Euler Products

Euler products are one of the central ideas of analytic number theory. They express infinite sums over integers as infinite products over primes.

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From Sums to Products

Euler products are one of the central ideas of analytic number theory. They express infinite sums over integers as infinite products over primes.

The basic principle comes from unique prime factorization. Since every positive integer decomposes uniquely into primes, multiplicative information about integers can often be reorganized prime by prime.

The simplest and most important example is the Euler product for the Riemann zeta function:

ζ(s)=∑n=1∞1ns=∏p11−p−s,Re⁡(s)>1. \zeta(s) = \sum_{n=1}^{\infty}\frac1{n^s} = \prod_p\frac1{1-p^{-s}}, \qquad \operatorname{Re}(s)>1.

This identity connects all positive integers on the left with all primes on the right.

Derivation of Euler’s Product Formula

For each prime pp,

1+p−s+p−2s+⋯=11−p−s 1+p^{-s}+p^{-2s}+\cdots = \frac1{1-p^{-s}}

when

∣p−s∣<1. |p^{-s}|<1.

Now multiply these geometric series over all primes:

∏p(1+p−s+p−2s+⋯ ). \prod_p \left( 1+p^{-s}+p^{-2s}+\cdots \right).

Expanding the product formally produces terms

p1−α1sp2−α2s⋯pr−αrs=1(p1α1p2α2⋯prαr)s. p_1^{-\alpha_1 s}p_2^{-\alpha_2 s}\cdots p_r^{-\alpha_r s} = \frac1{(p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_r^{\alpha_r})^s}.

By unique prime factorization, every positive integer appears exactly once.

Therefore the product becomes

∑n=1∞1ns. \sum_{n=1}^{\infty}\frac1{n^s}.

Hence

ζ(s)=∏p11−p−s. \zeta(s) = \prod_p\frac1{1-p^{-s}}.

Why Absolute Convergence Matters

The derivation above requires rearranging infinitely many terms. Such rearrangements are justified when the series converges absolutely.

For the zeta function,

∑n=1∞1ns \sum_{n=1}^{\infty}\frac1{n^s}

converges absolutely when

Re⁡(s)>1. \operatorname{Re}(s)>1.

In this region, the Euler product is valid.

Outside this region, the zeta function can still be extended analytically, but the infinite product no longer converges absolutely.

Prime Factorization Encoded Analytically

Euler products are analytic expressions of prime factorization.

The left side of the zeta identity sums over all integers. The right side factors the same information prime by prime.

Thus multiplication of integers becomes multiplication of local prime contributions.

This is one of the deepest structural ideas in number theory.

General Euler Products

Suppose ff is a multiplicative arithmetic function. Then its Dirichlet series has an Euler product:

∑n=1∞f(n)ns=∏p(1+f(p)ps+f(p2)p2s+⋯ ). \sum_{n=1}^{\infty}\frac{f(n)}{n^s} = \prod_p \left( 1+\frac{f(p)}{p^s} +\frac{f(p^2)}{p^{2s}} +\cdots \right).

The local factor at each prime records the values of ff on powers of that prime.

This identity follows from the same reasoning as the zeta function product: expanding the infinite product reproduces all prime factorizations exactly once.

Completely Multiplicative Functions

If ff is completely multiplicative, then

f(pα)=f(p)α. f(p^\alpha)=f(p)^\alpha.

The local factor becomes a geometric series:

1+f(p)ps+f(p)2p2s+⋯=11−f(p)p−s. 1+\frac{f(p)}{p^s} +\frac{f(p)^2}{p^{2s}} +\cdots = \frac1{1-f(p)p^{-s}}.

Therefore

∑n=1∞f(n)ns=∏p11−f(p)p−s. \sum_{n=1}^{\infty}\frac{f(n)}{n^s} = \prod_p \frac1{1-f(p)p^{-s}}.

For example, the Liouville function satisfies

λ(p)=−1, \lambda(p)=-1,

so

∑n=1∞λ(n)ns=∏p11+p−s. \sum_{n=1}^{\infty}\frac{\lambda(n)}{n^s} = \prod_p\frac1{1+p^{-s}}.

Using algebra,

11+p−s=1−p−s1−p−2s, \frac1{1+p^{-s}} = \frac{1-p^{-s}}{1-p^{-2s}},

which gives

∑n=1∞λ(n)ns=ζ(2s)ζ(s). \sum_{n=1}^{\infty}\frac{\lambda(n)}{n^s} = \frac{\zeta(2s)}{\zeta(s)}.

Möbius Function Euler Product

Since

μ(pα)=0 \mu(p^\alpha)=0

for

α≥2, \alpha\ge2,

the local factor for the Möbius function is

1−1ps. 1-\frac1{p^s}.

Therefore

∑n=1∞μ(n)ns=∏p(1−1ps). \sum_{n=1}^{\infty}\frac{\mu(n)}{n^s} = \prod_p\left(1-\frac1{p^s}\right).

But Euler’s product for ζ(s)\zeta(s) gives

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

Hence

∑n=1∞μ(n)ns=1ζ(s). \sum_{n=1}^{\infty}\frac{\mu(n)}{n^s} = \frac1{\zeta(s)}.

This identity is fundamental because zeros of ζ(s)\zeta(s) govern cancellation in the Möbius function.

Euler Products and Prime Distribution

Euler products reveal that primes control analytic behavior.

For example, taking logarithms of the zeta Euler product gives

log⁡ζ(s)=−∑plog⁡(1−p−s). \log\zeta(s) = -\sum_p\log(1-p^{-s}).

Using the approximation

−log⁡(1−x)≈x, -\log(1-x)\approx x,

one obtains

log⁡ζ(s)≈∑p1ps. \log\zeta(s) \approx \sum_p\frac1{p^s}.

Thus divergence or convergence of prime reciprocal sums is reflected in analytic behavior of ζ(s)\zeta(s).

In particular, the divergence of

∑p1p \sum_p\frac1p

corresponds to the singularity of ζ(s)\zeta(s) at

s=1. s=1.

Local-to-Global Principle

Euler products embody a local-to-global philosophy.

Each prime contributes a local factor. Multiplying all local factors reconstructs global arithmetic information.

This idea extends far beyond elementary number theory. Modern number theory studies LL-functions attached to algebraic objects such as characters, elliptic curves, modular forms, and number fields. Each such LL-function has an Euler product reflecting local behavior at primes.

Thus Euler products provide a universal language connecting arithmetic objects with analytic functions.

Role in Number Theory

Euler products are analytic manifestations of unique prime factorization.

They transform multiplicative arithmetic into infinite products indexed by primes. Through these products, prime distribution influences analytic behavior such as poles, zeros, and growth.

This connection lies at the heart of analytic number theory. The zeta function, Dirichlet LL-functions, modular forms, and many modern theories are built upon Euler products and the arithmetic structure they encode.