Jacques Hadamard

Jacques Hadamard (1865–1963) was a French mathematician whose work spanned number theory, complex analysis, and partial differential equations. He proved the prime number theorem independently of Charles de la Vallée Poussin.

Charles de la Vallée Poussin: A Belgian mathematician who independently proved the prime number theorem in 1896. He reached the same theorem as Hadamard independently, using a different proof.
Prime number theorem: A theorem stating that the number of primes not exceeding x is asymptotic to x divided by its natural logarithm. Hadamard proved it in 1896 by applying complex analysis to the Riemann zeta function.
Riemann zeta function: A complex function defined initially by a series over positive integers and extended by analytic continuation. Its zeros and pole encode the prime distribution that Hadamard studied.
Hadamard factorization theorem: A theorem expressing an entire function of finite order through its zeros and an exponential factor. It gives a structural account of entire functions that supported analytic number theory.
Elementary proof of the prime number theorem: A proof of the prime number theorem that avoids complex analysis and analytic properties of the zeta function. Hadamard’s 1896 proof was analytic, unlike the later elementary proofs of Erdős and Selberg.
Émile Borel: A French mathematician who helped establish measure theory and probability theory. Borel was Hadamard’s contemporary in the Paris mathematical community and a fellow member of the French Academy.
Hadamard's inequality: An upper bound on the absolute determinant of a matrix in terms of the Euclidean lengths of its rows or columns. This determinant bound is one of Hadamard’s best-known results in linear algebra.
Complex analysis: The study of functions of a complex variable, including their analytic structure and contour integrals. Hadamard’s proof of the prime number theorem relies on properties of analytic functions.
Hadamard's three-circle theorem: A theorem stating that the logarithm of the maximum modulus of an analytic function is convex in the logarithm of radius. It provides a growth estimate used in complex analysis and the study of entire functions.
Bernhard Riemann: A German mathematician whose work on complex analysis and the zeta function shaped modern number theory. Riemann supplied the zeta-function framework that Hadamard later used to prove the prime number theorem.

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