Euler product

An infinite product over primes that represents a Dirichlet series when its terms satisfy suitable multiplicative conditions and the product converges.

Multiplicative function: An arithmetic function satisfying f(mn)=f(m)f(n) whenever m and n are coprime. Multiplicative coefficients allow a Dirichlet series to split into prime-indexed factors.
Riemann zeta function: The function defined for complex s with real part greater than one by the series sum of n^{-s}. Its product over primes is the canonical Euler product.
Leonhard Euler: An eighteenth-century Swiss mathematician whose work shaped number theory, analysis, and mathematical notation. Euler introduced the prime product identity for the zeta function.
Prime number theorem: The theorem that the number of primes at most x is asymptotic to x divided by its natural logarithm. Analytic properties of the zeta function’s Euler product inform proofs about prime density.
Additive function: An arithmetic function satisfying f(mn)=f(m)+f(n) whenever m and n are coprime. Additive coefficients generally do not split a Dirichlet series into independent prime factors.
Unique factorization theorem: The theorem that integers greater than one factor uniquely into primes, up to ordering. It lets products over primes encode contributions from all positive integers.
Dirichlet L-function: A complex function defined by a Dirichlet series whose coefficients are values of a Dirichlet character. Its Euler product incorporates the character values at each prime.
Basel problem: The problem of finding the sum of the reciprocals of the squares of positive integers. Euler’s solution used the zeta function whose prime product became a landmark result.
Riemann hypothesis: The conjecture that every nontrivial zero of the Riemann zeta function has real part one-half. The product’s breakdown at zeros and its analytic continuation connect it to prime-counting error bounds.
Euler product formula for the gamma function: An infinite-product representation of the gamma function involving factors indexed by positive integers. It is called an Euler product but is not a product indexed by primes.

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