TheoremProved

Euler product formula for ζ(s)

Statement

For $\mathrm{Re}(s)>1$ , the Riemann zeta function satisfies $\zeta(s)=\displaystyle\sum_{n=1}^\infty \frac{1}{n^s} = \prod_p \frac{1}{1-p^{-s}}$ , the product ranging over all primes $p$ .

Why is it true?

Because every positive integer factors uniquely into primes, summing $1/n^s$ over all $n$ is the same as choosing, for each prime independently, how many times it appears in $n$ — exactly what the geometric series $\sum_{k\ge0} p^{-ks} = (1-p^{-s})^{-1}$ encodes for each prime.

Proof sketch

Expand each factor $(1-p^{-s})^{-1}=\sum_{k\ge0}p^{-ks}$ as a geometric series (valid for $\mathrm{Re}(s)>1$ ). Multiplying these series over all primes $p\le N$ and using unique factorization, every integer $n\le N$ appears exactly once as a term $n^{-s}$ in the expansion; the remaining terms (from integers with a prime factor $>N$ , or products beyond $N$ ) form a tail that vanishes as $N\to\infty$ , giving $\zeta(s)=\prod_p(1-p^{-s})^{-1}$ .

Stated by

Leonhard Euler

Proved by

proved 1737Leonhard Euler

Topics that use this theorem

The Riemann zeta function and the Riemann hypothesis

Related theorems

Infinitude of primes (Euclid)Prime number theorem Dirichlet's theorem on primes in arithmetic progressions

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Leonhard Euler (1737). Variae observationes circa series infinitas
Tom M. Apostol (1976). Introduction to Analytic Number Theory · DOI:10.1007/978-1-4757-5579-4