Johann Peter Gustav Lejeune Dirichlet (German: [ləˈʒœn diʀiˈkleː];[1] 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a function.

Although his surname is Lejeune Dirichlet, he is commonly referred to as just Dirichlet, in particular for results named after him. Gustav Lejeune Dirichlet was born on 13 February 1805 in Düren University of Paris, learning mathematics from Hachette among others, while undertaking private study of Gauss's Disquisitiones Arithmeticae, a book he kept close for his entire life. In 1823 he was recommended to General Maximilien Foy, who hired him as a private tutor to teach his children German, the wage finally allowing Dirichlet to become independent from his parents' financial support.[3]

His first original research, comprising part of a proof of Fermat's Last Theorem for the case n = 5, brought him immediate fame, being the first advance in the theorem since Fermat's own proof of the case n = 4 and Euler's proof for n = 3. Adrien-Marie Legendre, one of the referees, soon completed the proof for this case; Dirichlet completed his own proof a short time after Legendre, and a few years later produced a full proof for the case n = 14.[4] In June 1825 he was accepted to lecture on his partial proof for the case n = 5 at the French Academy of Sciences, an exceptional feat for a 20-year-old student with no degree.[2] His lecture at the Academy had also put Dirichlet in close contact with Fourier and Poisson, who raised his interest in theoretical physics, especially Fourier's analytic theory of heat.After Dirichlet's move to Berlin, Humboldt introduced him to the great salons held by the banker Abraham Mendelssohn Bartholdy and his family. Their house was a weekly gathering point for Berlin artists and scientists, including Abraham's children Felix and Fanny Mendelssohn, both outstanding musicians, and the painter Wilhelm Hensel (Fanny's husband). Dirichlet showed great interest in Abraham's daughter Rebecka, whom he married in 1832.

Rebecka Henriette Lejeune Dirichlet (née Rebecka Mendelssohn; 11 April 1811 – 1 December 1858) was a granddaughter of Moses Mendelssohn and the youngest sister of Felix Mendelssohn and Fanny Mendelssohn.[6][7] Rebecka was born in Hamburg.[8] In 1816 her parents arranged for her to be baptised at which point she took the names Rebecka Henriette Mendelssohn Bartholdy.[9] She became a part of the notable salon of her parents, Abraham Mendelssohn and his wife Lea, having social contacts with the important musicians, artists and scientists in a highly creative period of German intellectual life. In 1829 she sang a small role in the premiere, given at the Mendelssohn house, of Felix's Singspiel Die Heimkehr aus der Fremde. n 1833 their first son, Walter, was born. She died in Göttingen in 1858.