Math 2.0

Number theory Analytic number theory Theorem

Euler product formula

Builds on Prime numberFundamental theorem of arithmetic

Statement

Let ss be a complex number with real part Re⁡(s)>1\operatorname{Re}(s) > 1, and let ζ(s)=∑n=1∞n−s\zeta(s) = \sum_{n=1}^{\infty} n^{-s} be the Riemann zeta function. Then ζ(s)=∏p11−p−s,\zeta(s) = \prod_{p} \frac{1}{1 - p^{-s}}, where the product runs over all prime numbers pp and converges absolutely.

Remarks

The proof expands each factor as a geometric series 1+p−s+p−2s+⋯1 + p^{-s} + p^{-2s} + \cdots. The fundamental theorem of arithmetic shows that the product of these series contains each term n−sn^{-s} exactly one time. The same method gives a product formula ∑nf(n)n−s=∏p∑k≥0f(pk)p−ks\sum_n f(n) n^{-s} = \prod_p \sum_{k \ge 0} f(p^k) p^{-ks} for each multiplicative function ff when the series converges absolutely. Since ζ(s)→∞\zeta(s) \to \infty as s→1+s \to 1^+, the formula gives a proof of Euclid's theorem.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Divisibility, definition, no evidence levelDivisibilitydefinitionPrime number, definition, no evidence levelPrime numberdefinitionEuclid's lemma, statement, no evidence levelEuclid's lemmastatementExistence of a prime divisor, statement, no evidence levelExistence of a prime …statementPrime factorization, definition, no evidence levelPrime factorizationdefinitionExistence of prime factorization, statement, no evidence levelExistence of prime fa…statementFundamental theorem of arithmetic, statement, no evidence levelFundamental theorem o…statementEuler product formula, statement, no evidence levelEuler product formulastatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Divisibility definition
  2. Prime number definition
  3. Euclid's lemma statement
  4. Existence of a prime divisor statement
  5. Prime factorization definition
  6. Existence of prime factorization statement
  7. Fundamental theorem of arithmetic statement
  8. Euler product formula statement

Where this leads

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Further reading

  1. Web https://en.wikipedia.org/wiki/Euler_product, Examples
About this page
Tag0C3PN4075R
URIm2://0C3PN4075R
Kindstatement, theorem
WritingW0 stub
MSC11M06
StatusThis copy of the site has no build data. Run m2 status 0C3PN4075R.
UsesPrime number, Fundamental theorem of arithmetic

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