On the one hand, the year 2026 marks the bicentenary of the birth of Bernhard Riemann (1826–1866); on the other hand, this anniversary has been–and continues to be–commemorated on numerous other occasions. Consequently, the present contribution is devoted not exclusively to Riemann, but also to a mathematician who was closely associated with him–particularly through the naming of a significant theorem.
Gustav Roch was born in Dresden on December 9, 1839; he initially studied chemistry there at the Royal Polytechnical School, but switched to mathematics in 1859, which he studied in Leipzig, Göttingen, and Berlin. In 1862, he received his doctorate in mathematics from Leipzig with a dissertation “Über die Darstellung von Functionen dreier Variablen durch Potentialausdrücke“. The reviewers were the physicist Wilhelm Hankel (1814–1899) and the mathematician Wilhelm Scheibner (1826–1908). From Leipzig, Roch moved to the University of Halle, where he became “Privatdozent” in 1863 and associate professor in 1866. On November 21 of that same year, he died of tuberculosis.
Thanks to a scholarship, Roch had been able to go to Göttingen, where he stayed from the summer semester 1861 to the summer semester 1862, attending in particular the lectures of Riemann, who deeply impressed him. In 1857, Riemann had published his groundbreaking work “Theorie der Abel’schen Functionen” in Volume 54 of Leopold August Crelle's (1780–1855) Journal für die reine und angewandte Mathematik. (This journal, too, came into existence exactly two hundred years ago.) In § 3 of this article, Riemann provides a topological definition of the genus p of what is today referred to as a Riemann surface, and demonstrates in § 5 that the general expression for a function possessing simple poles at m distinct points of the Riemann surface depends on at least m - p + 1 arbitrary parameters. Put differently: For the number N of arbitrary parameters for this class of functions one has the inequality N – (m – p + 1) ≥ 0, which is known as Riemann's inequality.
Roch undertook the task of endowing the number N – (m – p + 1) with a mathematical significance. In 1865, he published a paper spanning only five pages titled “Ueber die Anzahl der willkürlichen Constanten in algebraischen Functionen”–also in the Journal für die reine und angewandte Mathematik–in which he demonstrated that Riemann's inequality could be turned into an equality by explaining the missing term. Following a suggestion made in 1874 by Alexander (from 1897: von) Brill (1842–1935) and Max Noether (1844–1921), this equality is referred to as the Riemann-Roch Theorem.
Nowadays, when formulating this theorem for Riemann surfaces, one employs the language of divisors; when stating it for algebraic curves, one utilizes the language of cohomology spaces with values in the sheaf of regular functions. Neither of these two terminologies was available to Roch–at least not explicitly. We will examine how he managed to prove and formulate the theorem solely by using algebraic functions, integrals of second kind, and linear relations between parameters.