Jacques Hadamard (1865\u20131963) was a French mathematician whose most important result was a proof of the prime number theorem in 1896. Today his name also appears in the Hadamard gate used in quantum circuits, in Hadamard matrices and in the Walsh\u2013Hadamard transform, which is why many developers first meet him through code rather than through number theory.<\/p>\n
This article focuses on what can be stated with confidence from the sources: his life, his career and his best-documented contributions. It also corrects several errors that circulate online, including some from an earlier version of this page.<\/p>\n
The short version, drawn mainly from the MacTutor biography of Jacques Hadamard<\/a> and the Dictionary of Scientific Biography entry<\/a>:<\/p>\n Hadamard’s father, Am\u00e9d\u00e9e, taught at the Lyc\u00e9e Imp\u00e9rial in Versailles. According to MacTutor<\/a>, the family moved to Paris when Jacques was three and lived through the Prussian siege of Paris in 1870.<\/p>\n He was not an obvious mathematical prodigy. MacTutor notes that he was weak in arithmetic at first and excelled at Greek and Latin. That changed during his schooling: in 1884 he placed first in the entrance examinations for both the \u00c9cole Polytechnique and the \u00c9cole Normale Sup\u00e9rieure, and chose the ENS. His fellow students included Pierre Duhem and Paul Painlev\u00e9.<\/p>\n He studied at the ENS from 1884 to 1888 and then taught at the Lyc\u00e9e Buffon from 1890 to 1893, according to the DSB entry<\/a>. In 1892 he received his doctorate for a thesis on functions given by their Taylor series, Essai sur l’\u00e9tude des fonctions donn\u00e9es par leur d\u00e9veloppement de Taylor<\/em>, as recorded by the Fondation Hadamard<\/a>. Wikipedia<\/a> lists his doctoral advisors as \u00c9mile Picard and Jules Tannery.<\/p>\n The thesis turned out to matter far beyond its original subject. MacTutor reports that Hadamard realised his results applied to zeta functions, and his paper on entire functions and zeta functions won first prize in the 1892 Grand Prix des Sciences Math\u00e9matiques. That work laid the groundwork for the result he is best known for.<\/p>\n The prime number theorem describes how prime numbers thin out: the number of primes up to a large number x<\/em> is approximately x<\/em> divided by the natural logarithm of x<\/em>. Mathematicians had suspected this for a long time, but a rigorous proof was missing.<\/p>\n In 1896 Jacques Hadamard proved it using complex function theory, the same toolkit he had developed for his work on entire functions and zeta functions. MacTutor calls the prime number theorem his “most important result”.<\/p>\n In the same year, the Belgian mathematician Charles-Jean de la Vall\u00e9e Poussin also proved the theorem. As Wikipedia’s Hadamard article<\/a> notes, the two proofs were found independently. They did not collaborate, and the phrase “in collaboration with”, which appears in some popular accounts, is wrong. Both men are credited with the proof, and both proofs relied on complex analysis.<\/p>\n After his doctorate Hadamard moved through several of France’s leading institutions. The DSB<\/a> gives the following sequence:<\/p>\n\n
Early life, the \u00c9cole Normale Sup\u00e9rieure and the 1892 thesis<\/h2>\n
The 1892 Grand Prix<\/h3>\n
Jacques Hadamard and the 1896 prime number theorem<\/h2>\n
What “independently of de la Vall\u00e9e Poussin” means<\/h3>\n
Career: Bordeaux, the Sorbonne, the Coll\u00e8ge de France and Poincar\u00e9’s seat<\/h2>\n