# Bernhard Riemann > German mathematician (1826–1866) **Wikidata**: [Q42299](https://www.wikidata.org/wiki/Q42299) **Wikipedia**: [English](https://en.wikipedia.org/wiki/Bernhard_Riemann) **Source**: https://4ort.xyz/entity/bernhard-riemann ## Summary Bernhard Riemann was a German mathematician (1826–1866) who fundamentally transformed mathematics through his pioneering work in analysis, number theory, and differential geometry. He is best known for the Riemann hypothesis—one of mathematics' most famous unsolved conjectures—and for developing Riemannian geometry, which later became essential to Einstein's theory of general relativity. His brief life produced discoveries that continue to shape modern mathematics and theoretical physics. ## Biography - **Born**: September 17, 1826, in Breselenz, Kingdom of Hanover (now Germany) - **Nationality**: German (subject of the Kingdom of Hanover) - **Education**: University of Göttingen; additional institutions include Q20266330, Q1698887, Q120818394 - **Known for**: Riemann hypothesis, Riemann surfaces, Riemannian geometry, foundational work in mathematical analysis - **Employer(s)**: University of Göttingen - **Field(s)**: Mathematics, mathematical analysis, number theory, differential geometry, mathematical physics ## Contributions Riemann's contributions to mathematics are among the most influential in history: - **Riemann Hypothesis (1859)**: In his paper "On the Number of Primes Less Than a Given Magnitude," Riemann proposed the famous conjecture linking the distribution of prime numbers to the zeros of the Riemann zeta function. This remains one of mathematics' greatest unsolved problems, with a million-dollar prize awaiting its resolution. - **Riemann Surfaces**: He developed the theory of Riemann surfaces, one-dimensional complex manifolds that provide a natural domain for multi-valued functions like the complex logarithm and square root. This work founded the field of complex analysis in multiple variables. - **Riemannian Geometry**: His habilitation lecture "On the Hypotheses Which Lie at the Bases of Geometry" introduced the concept of Riemannian manifolds—smooth manifolds equipped with Riemannian metrics. This framework became the mathematical foundation for Einstein's general theory of relativity. - **Riemann Integral**: He formalized the concept of integration through what became known as the Riemann integral, establishing rigorous foundations for calculus. - **Riemann-Roch Theorem**: Developed fundamental results about holomorphic vector bundles on compact complex manifolds, with profound implications for algebraic geometry. - **Riemann Mapping Theorem**: Proved that any non-empty simply connected open proper subset of the complex plane is biholomorphic to the open unit disk. - **Cauchy-Riemann Equations**: Though named after Cauchy, Riemann contributed significantly to the understanding of these partial differential equations characterizing holomorphic functions. - **Other Contributions**: Developed the Riemann sum approximation technique, Riemann-Stieltjes integral, Riemann-Liouville integral, Riemann-Hurwitz formula, and Riemann invariants in conservation equations. ## FAQs ### What is Bernhard Riemann best known for? Bernhard Riemann is best known for the Riemann hypothesis, a conjecture about the distribution of prime numbers that remains unsolved since 1859, and for creating Riemannian geometry, which provided the mathematical framework for Einstein's general theory of relativity. ### Where did Bernhard Riemann study and work? Riemann studied at the University of Göttingen, one of Europe's leading mathematical centers. He completed his doctorate there in 1851 under Carl Friedrich Gauss and later became a professor at Göttingen, where he taught until his death in 1866. ### What is the Riemann hypothesis? The Riemann hypothesis, proposed in Riemann's 1859 paper "On the Number of Primes Less Than a Given Magnitude," conjectures that all non-trivial zeros of the Riemann zeta function have a real part of 1/2. This conjecture has profound implications for understanding the distribution of prime numbers. ### How did Riemann contribute to physics? Riemann's work in differential geometry, particularly his concept of curved spaces (Riemannian geometry), became the mathematical foundation for Einstein's general theory of relativity. His work on complex analysis and topology also influenced theoretical physics. ### When did Bernhard Riemann live? Bernhard Riemann was born on September 17, 1826, and died on July 20, 1866, at age 39. Despite his short life, his mathematical contributions revolutionized multiple fields. ## Why They Matter Riemann's influence on mathematics and physics cannot be overstated. His invention of Riemannian geometry provided the precise mathematical language needed to describe curved spaces, which Einstein later used to formulate general relativity—revolutionizing our understanding of gravity, space, and time. Without Riemann's work, modern physics would lack the mathematical framework for understanding black holes, the expanding universe, and the large-scale structure of cosmos. The Riemann hypothesis, though unproven, has driven enormous research in number theory for over 160 years. Its resolution would have profound implications for cryptography, computer science, and our fundamental understanding of prime numbers. The Clay Mathematics Institute has offered a one-million-dollar prize for its proof. In mathematical analysis, Riemann's rigorous formulation of integration became the foundation for measure theory and real analysis. His work on complex analysis, topology, and differential geometry created new fields of mathematical inquiry that continue to generate research today. Riemann's approach—linking geometry, analysis, and physics—exemplified a vision of mathematics as a unified whole that influenced subsequent generations of mathematicians and physicists. ## Notable For - **Riemann Hypothesis**: One of mathematics' seven "Millennium Prize Problems" with a $1 million reward for solution - **Riemannian Geometry**: Mathematical foundation for Einstein's general theory of relativity - **Foundational Work**: Established rigorous foundations for complex analysis and integration theory - **Royal Society Recognition**: Awarded the Royal Medal in 1866 - **Brief but Prolific Career**: Produced revolutionary work across multiple mathematical fields in just 15 years of research - **Influential Publications**: His 1859 paper on prime numbers remains among the most cited mathematical works in history ## Body ### Early Life and Education Bernhard Riemann was born on September 17, 1826, in Breselenz, a small village in the Kingdom of Hanover, in what is now Lower Saxony, Germany. His father, Friedrich Bernhard Riemann, was a Lutheran pastor who encouraged his son's intellectual development. Riemann showed exceptional mathematical abilities from an early age, though he was also deeply interested in theology and philosophy. Riemann began his formal education at the University of Göttingen in 1846, where he studied mathematics under Carl Friedrich Gauss, one of the greatest mathematicians in history. Gauss immediately recognized Riemann's extraordinary talent. Riemann completed his doctoral dissertation in 1851 at Göttingen, titled "Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen complexen Grösse" (Foundations for a General Theory of Functions of a Complex Variable). Gauss praised the work highly, noting it demonstrated "original, penetrating, and indeed glorious mathematical genius." ### Academic Career After completing his doctorate, Riemann remained at Göttingen, working toward his habilitation—the qualification required to become a university professor in Germany. His habilitation lecture in 1854, titled "Über die Hypothesen, welche der Geometrie zu Grunde liegen" (On the Hypotheses Which Lie at the Bases of Geometry), was a revolutionary presentation that laid the foundations of Riemannian geometry. Gauss, who attended the lecture, was reportedly astonished by the depth of Riemann's ideas. Riemann became a professor at Göttingen in 1857, succeeding his mentor Dirichlet. He held this position until his death in 1866. During this period, he produced his most significant work, including the papers on prime numbers, complex analysis, and the foundations of geometry. ### Mathematical Contributions **Complex Analysis and Riemann Surfaces**: Riemann's work on complex functions led him to develop the theory of Riemann surfaces—geometric objects that allow multi-valued functions to be treated as single-valued. This innovation resolved apparent paradoxes in complex analysis and established new connections between analysis, topology, and geometry. His Riemann mapping theorem demonstrated that any simply connected region in the complex plane (excluding the entire plane) can be mapped conformally to the unit disk. **Number Theory and the Riemann Zeta Function**: In 1859, Riemann was elected to the Berlin Academy of Sciences and submitted his only paper on number theory: "Über die Anzahl der Primzahlen unter einer gegebenen Grösse" (On the Number of Primes Less Than a Given Magnitude). This brief paper—its entire text would fit on about eight pages—introduced what became known as the Riemann zeta function and proposed the Riemann hypothesis. The connection between prime number distribution and the zeros of the zeta function that Riemann discovered has driven number theory research for over a century and a half. **Differential Geometry**: Riemann's habilitation lecture introduced the concept of n-dimensional curved spaces equipped with metrics. He showed that Euclidean geometry was just one possibility among infinitely many non-Euclidean geometries. This work, initially considered too abstract for practical application, became essential when Einstein sought the mathematical language to describe curved spacetime. **Real Analysis and Integration**: The Riemann integral provided rigorous foundations for calculus, addressing gaps in earlier work by Cauchy. His approach using Riemann sums became the standard undergraduate introduction to integration. He also developed the Riemann-Stieltjes integral, generalizing integration with respect to functions rather than measures. **Algebraic Geometry**: The Riemann-Roch theorem, developed independently by both Riemann and his student Gustav Roch, established fundamental relationships between the topology and algebraic geometry of Riemann surfaces. This theorem has extensions to higher-dimensional complex manifolds through the Hirzebruch-Riemann-Roch theorem and the Grothendieck-Riemann-Roch theorem. ### Influence and Legacy Riemann's influence extends across virtually every branch of modern mathematics and theoretical physics: - **Physics**: Riemannian geometry became the mathematical foundation for Einstein's general relativity (1915), fundamentally changing our understanding of gravity as spacetime curvature. Modern physics, from cosmology to particle theory, relies on Riemann's geometric framework. - **Mathematics**: His ideas spawned entire fields—complex analysis on Riemann surfaces, algebraic topology, and modern differential geometry. The Cauchy-Riemann equations remain fundamental to complex analysis, while the Riemann hypothesis drives research in analytic number theory. - **Influence on Other Mathematicians**: Riemann's work influenced subsequent generations including Henri Poincaré (who built on his topological ideas), David Hilbert, and the entire development of 20th-century mathematics. His approach of connecting different mathematical fields became a model for mathematical research. ### Personal Characteristics and Later Life Riemann was known for his gentle nature and deep concentration. He struggled with health problems throughout his life, likely suffering from tuberculosis. Despite his mathematical genius, he was also interested in philosophy and theology, reflecting the broad intellectual culture of 19th-century German academia. In 1862, Riemann married Elise Koch, a friend of his sister. They had one daughter, who survived him. Riemann spent his final years in Italy, seeking to improve his health, but died in Selasca, Italy, on July 20, 1866, at age 39. ### Recognition Riemann received significant recognition during his lifetime, including election to the Berlin Academy of Sciences in 1859 and the Royal Society in 1866. He was awarded the Royal Medal by the Royal Society in 1866, though he died before receiving the news. His collected works were published posthumously and continue to be studied for their profound insights. The lunar crater Riemann and the asteroid 4167 Riemann were named in his honor, testifying to his lasting impact on science. His mathematical concepts—Riemann surfaces, Riemannian geometry, the Riemann hypothesis—remain central to mathematics more than 150 years after his death. ## References 1. MacTutor History of Mathematics archive 2. BnF authorities 3. Integrated Authority File 4. Genealogics 5. datos.bne.es 6. Czech National Authority Database 7. Encyclopedia.com 8. Find a Grave 9. Complete List of Royal Society Fellows 1660-2007 10. [Source](http://www.genealogy.ams.org/id.php?id=18232) 11. Mathematics Genealogy Project 12. International Standard Name Identifier 13. Virtual International Authority File 14. CiNii Research 15. [Bernhard Riemann 1826–1866: Turning Points in the Conception of Mathematics](https://books.google.de/books?id=uTFNAAAAQBAJ&pg=PA23&lpg=PA23&dq=riemann+royal+society&source=bl&ots=Vrjf2jy0y8&sig=COclIOdWvwlnHwOUbyVHLAT-6IA&hl=en&sa=X&ved=0ahUKEwiCss2xs6zWAhUK1hQKHVDCBeoQ6AEIXzAJ#v=onepage&q=riemann%20royal%20society&f=false) 16. [Bernhard Riemann 1826–1866: Turning Points in the Conception of Mathematics](https://books.google.de/books?id=uTFNAAAAQBAJ&lpg=PA23&ots=Vrjf2jy0y8&dq=riemann%20royal%20society&pg=PA23#v=onepage&q=riemann%20royal%20society&f=false) 17. [Bernhard Riemann 1826–1866: Turning Points in the Conception of Mathematicsen](https://books.google.de/books?id=uTFNAAAAQBAJ&lpg=PA23&ots=Vrjf2jy0y8&dq=riemann%20royal%20society&pg=PA23#v=onepage&q=riemann%20royal%20society&f=false) 18. [Source](https://kalliope-verbund.info/DE-611-BF-3876) 19. [Source](https://kalliope-verbund.info/DE-611-BF-61672) 20. SNAC 21. Brockhaus Enzyklopädie 22. Great Norwegian Encyclopedia 23. Gran Enciclopèdia Catalana 24. Croatian Encyclopedia 25. Internet Philosophy Ontology project 26. Freebase Data Dumps. 2013 27. [Source](http://digitale.beic.it/primo_library/libweb/action/search.do?fn=search&vid=BEIC&vl%283134987UI0%29=creator&vl%28freeText0%29=Riemann%20Bernhard) 28. CONOR.SI 29. Autoritats UB 30. BBC Things 31. 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