Math 2.0

Number theory Theorem

Euler product for multiplicative functions

Builds on Multiplicative functionPrime numberAbsolute convergenceFundamental theorem of arithmeticInfinite product

Statement

Let ff be a multiplicative function from the positive integers to ℂ\C such that ∑n=1∞f(n)\sum_{n=1}^{\infty} f(n) converges absolutely. Then for each prime pp the series ∑k=0∞f(pk)\sum_{k=0}^{\infty} f(p^k) converges absolutely, and ∑n=1∞f(n)=lim⁡x→∞∏p≤x∑k=0∞f(pk),\sum_{n=1}^{\infty} f(n) = \lim_{x \to \infty} \prod_{p \le x} \sum_{k=0}^{\infty} f(p^k), where the product runs over the primes p≤xp \le x. If ff is completely multiplicative, then ∣f(p)∣<1\abs{f(p)} < 1 for each prime pp, and the factor for pp equals 1/(1−f(p))1/(1 - f(p)).

Remarks

For finite xx, multiplying out the finitely many absolutely convergent factors gives ∑f(n)\sum f(n) over all nn whose prime factors are at most xx. Here one uses that ff is multiplicative and the fundamental theorem of arithmetic. Every missing nn is larger than xx, so the difference from the full sum is at most ∑n>x∣f(n)∣\sum_{n > x} \abs{f(n)}, which tends to 00. With f(n)=n−sf(n) = n^{-s} and Re⁡(s)>1\operatorname{Re}(s) > 1 this is the Euler product formula for the zeta function. With f(n)=μ(n)n−sf(n) = \mu(n) n^{-s}, where μ\mu is the Möbius function, it gives ∑μ(n)n−s=∏p(1−p−s)\sum \mu(n) n^{-s} = \prod_p (1 - p^{-s}).

Going deeper

For a student

Euler products are the starting point of analytic number theory. The zeta function, Dirichlet LL-functions and many other Dirichlet series have them in their half-plane of absolute convergence. From the product one reads off facts about primes and zeros. For example, the zeta function has no zeros where its Euler product converges, since a convergent product of nonzero factors is nonzero.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

Show the path as a graph
Arithmetic function, definition, no evidence levelArithmetic functiondefinitionComplex numbers, object, no evidence levelComplex numbersobjectDivisibility, definition, no evidence levelDivisibilitydefinitionSequence, definition, no evidence levelSequencedefinitionGreatest common divisor, definition, no evidence levelGreatest common divis…definitionInfinite product, definition, no evidence levelInfinite productdefinitionPrime number, definition, no evidence levelPrime numberdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionCoprime integers, definition, no evidence levelCoprime integersdefinitionEuclid'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 factorizationdefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionMultiplicative function, definition, no evidence levelMultiplicative functi…definitionExistence of prime factorization, statement, no evidence levelExistence of prime fa…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionFundamental theorem of arithmetic, statement, no evidence levelFundamental theorem o…statementConvergent series, definition, no evidence levelConvergent seriesdefinitionAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionEuler product for multiplicative functions, statement, no evidence levelEuler product for mul…statement
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.
  • no evidence level
  1. Arithmetic function definition
  2. Complex numbers object
  3. Divisibility definition
  4. Sequence definition
  5. Greatest common divisor definition
  6. Infinite product definition
  7. Prime number definition
  8. Real sequence definition
  9. Coprime integers definition
  10. Euclid's lemma statement
  11. Existence of a prime divisor statement
  12. Prime factorization definition
  13. Limit of a sequence definition
  14. Series definition
  15. Multiplicative function definition
  16. Existence of prime factorization statement
  17. Convergent sequence definition
  18. Partial sum definition
  19. Fundamental theorem of arithmetic statement
  20. Convergent series definition
  21. Absolute convergence definition
  22. Euler product for multiplicative functions statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Euler_product
About this page
TagBVA6XT55CS
URIm2://BVA6XT55CS
Kindstatement, theorem
WritingW0 stub
MSC11A25, 11M06, 40A20
StatusThis copy of the site has no build data. Run m2 status BVA6XT55CS.
AboutInfinite product
UsesMultiplicative function, Prime number, Absolute convergence, Fundamental theorem of arithmetic, Infinite product
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Changed2 November 2026

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