Hadamard factorization theorem

The Hadamard factorization theorem expresses an entire function of finite order as a canonical product over its zeros multiplied by an exponential of a polynomial.

Weierstrass factorization theorem: A theorem representing entire functions as products built from their zeros and a zero-free factor. Hadamard’s result strengthens this general factorization by controlling the product and factor through finite order.
Entire function: A complex-valued function holomorphic at every point of the complex plane. The theorem applies to entire functions, whose zeros can be listed with multiplicities.
Riemann xi function: An entire function built from the Riemann zeta function whose zeros correspond to its nontrivial zeros. Its Hadamard product translates the nontrivial zeta zeros into identities involving sums over zeros.
Mittag-Leffler theorem: A theorem constructing meromorphic functions with prescribed poles and principal parts. It prescribes polar behavior, whereas Hadamard factorization represents entire functions through zeros.
Canonical product: An infinite product whose primary factors make it converge while encoding a prescribed sequence of zeros. The theorem’s zero-dependent part is a canonical product with genus constrained by growth.
Zero of a holomorphic function: A point at which a holomorphic function takes the value zero, counted with its multiplicity. The canonical product records every nonzero zero with its multiplicity.
Riemann zeta function: A meromorphic function defined initially by the Dirichlet series sum n⁻ˢ and continued to the complex plane. Factorization of its completed form yields relations between its zeros and special values.
Hadamard factorization theorem for Dirichlet series: A theorem describing factorization properties of certain Dirichlet series under analytic hypotheses. The similarly named result concerns Dirichlet series, not entire functions and their canonical products.
Weierstrass primary factor: A factor of the form E_p(w)=(1-w) exp(w+w²/2+…+w^p/p), used to ensure convergence of infinite products. These factors regularize the product over zeros in the theorem’s representation.
Maximum modulus principle: A theorem stating that a nonconstant holomorphic function cannot attain a local maximum of its modulus inside its domain. Maximum-modulus growth estimates help define finite order and constrain the factorization.

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Jacques HadamardBroader topic: It gives a structural account of entire functions that supported analytic number theory.