Riemann Hypothesis/Biography of Riemann

Bernhard Riemann was born in the Kingdom of Hanover, in modern Germany, in 1826. From an early age, Riemann directed much of his interest towards theology and philology. While he excelled in mathematics, he was rather timid and shy, and neglected to display much of his ability. When Riemann attended the University of Göttingen, he initially aspired to study Theology. At university, Riemann met Gauss, who advised him to give up his theological studies, and pursue mathematics. He later transferred to the University of Berlin to read mathematics, where several notable mathematicians, including Steiner, Jacobi and Dirichlet (from whom he would borrow concepts for his later studies). Riemann died from tuberculosis on a trip to Italy in 1866. Despite his early death, Riemann left a significant legacy that can still be seen in mathematics to this day.

Riemann's academic work primarily concerned analysis, number theory and differential geometry. He is credited with his contribution of the Riemann integral, the first formal, rigorous, definition of an integral, Riemann surfaces and Riemannian geometry, the latter being later used by Einstein as part of his theory of General Relativity. This piece concerns Riemann's research in real analysis, specifically prime numbers, where he explored the distribution of prime numbers, later proposing the Riemann Hypothesis in 1859.

Riemann published his paper, "On the Number of Primes Less Than a Given Magnitude" (Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse) in 1859, in the journal "Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin". The paper's primary focus is the prime-counting functions, $$\pi$$, which, as the title of the paper suggests, gives the number of primes less than a given argument. Among other things, Riemann introduced definitions, proofs and methods that would later prove to be invaluable in further research, including Fourier inversion, analytical continuation, and Contour integration.