Bernhard Riemann

1. Bernhard Riemann By: Aleena Nie

2. Early Life • born on September 17th, 1826 in Breselenz, Germany • Father: Friedrich Bernhard Riemann was a Lutheran minister • Mother: Charlotte Ebell had passed away shortly after giving birth to the last of her six children Breselenz, Germany

3. Early Education • Father home schooled him until he was ten • teacher from the local school, Schulz, helped him with his education • in 1840 Riemann directly transferred into the third class at the Lyceum in Hannover (which is in present day Germany), • in 1842 he moved to the Johanneum Gymnasium in the University of Lüneburg

4. Attitude as a Student • He was average • But some of the teachers (mostly later on in his higher education) noticed that he was extremely gifted in math and that his genius would go far • sometimes, the professor would let Riemann study math textbooks from his own personal library

5. Higher Education • in 1846 Bernhard Riemann enrolled at University of Göttingen, with his fathers encouragement and direction to study theology • attended some mathematical lectures and finally asked his father if he could study mathematics

6. Professors Riemann Studied Under Higher Education continued • studied under many intelligent professors, two of which were Moritz Stern and Gauss • moved from the University of Göttingen to Berlin University in 1847 where he now studied under Steiner, Jacobi, Dirichlet, and Einstein

7. Influence • Einstein and Riemann discussed using complex variables in elliptic function theory • however, the main person that • influenced Riemann was Dirichlet. It was said…. “Riemann was bound to Dirichlet by the strong inner sympathy of a like mode of thought. Dirichlet loved to make things clear to himself in an intuitive substrate; along with this he would give acute, logical analyses of foundational questions and would avoid long computations as much as possible. His manner suited Riemann, who adopted it and worked according to Dirichlet's methods.”

8. Theory of Higher Dimensions • “In 1853, Gauss asked his student Riemann to prepare a Habilitationsschrift on the foundations of geometry. Over many months, Riemann developed his theory of higher dimensions. When he finally delivered his lecture at Göttingen in 1854, the mathematical public received it with enthusiasm, and it is one of the most important works in geometry.

9. Theory of Higher Dimensions Continued • It was titled Über die Hypothesen welche der Geometrie zu Grunde liegen (loosely: "On the foundations of geometry"; more precisely, "On the hypotheses which underlie geometry"), and was published in 1868. The subject founded by this work is Riemannian geometry. Riemann found the correct way to extend into n dimensions the differential geometry of surfaces, which Gauss himself proved in his theorema egregium.

10. Theory of Higher Dimensions Concluded • The fundamental object is called the Riemann curvature tensor. For the surface case, this can be reduced to a number (scalar), positive, negative or zero; the non-zero and constant cases being models of the known non-Euclidean geometries.” (“Bernhard Riemann”)

11. Zeta Function • in 1859 Riemann was elected to the Berlin Academy of Sciences and newly elected members of this must present their most recent research • “Riemann sent a report on On the number of primes less than a given magnitude another of his great masterpieces which were to change the direction of mathematical research in a most significant way. In it Riemann examined the zeta function: • ζ(s) = ∑ (1/ns) = ∏ (1 - p-s)-1

12. Zeta Function Continued • which had already been considered by Euler. Here the sum is over all natural numbers n while the product is over all prime numbers. Riemann considered a very different question to the one Euler had considered, for he looked at the zeta function as a complex function rather than a real one. Except for a few trivial exceptions, the roots of ζ(s) all lie between 0 and 1.

13. Zeta Function Continued • In the paper he stated that the zeta function had infinitely many nontrivial roots and that it seemed probable that they all have real part 1/2. This is the famous Riemann hypothesis which remains today one of the most important of the unsolved problems of mathematics.

14. Zeta function Conclusion • Riemann studied the convergence of the series representation of the zeta function and found a functional equation for the zeta function. The main purpose of the paper was to give estimates for the number of primes less than a given number. Many of the results which Riemann obtained were later proved by Hadamard and de la Vallee Poussin.” (“Riemann biography”)

15. Other Discoveries • also known for many other things such as Riemann hypothesis, Riemann integral, Riemann sphere, Riemann differential equation, Riemann mapping theorem, Riemann curvature tensor, Riemann sum, Riemann-Stieltjes integral, Cauchy-Riemann equations, Riemann-Hurwitz formula, Riemann-Lebesgue lemma, Riemann-von Mangoldt formula, Riemann problem, Riemann series theorem, Hirzebruch-Riemann-Roch theorem, Riemann Xi function….and many others.

16. Math Today • His zeta function is used in statistics as well as some scientific formulas. Even today there is a one million dollar cash prize (as seen in New York Times in 2002) for any person that can find a proof of Riemann’s hypothesis.

17. Bibliography • Alchian, Armen A., Glenn James, Edwin F. Beckenbach, Clifford Bell, Homer V. Craig, Robert C. James, Aristotle D. Michal, and Ivan S. Sokolnikoff. Mathematics Dictionary. 5th ed. New York: Chapman & Hall, 1992. • "Bernhard Riemann -." Wikipedia, the free encyclopedia. 20 May 2009 <http://en.wikipedia.org/wiki/ Bernhard_Riemann>. • Considine, Douglas M., and Glenn D. Considine. Scientific Encyclopedia. 8th ed. New York: Van Nostrana's Reinhold, 1995. • "Riemann biography." GAP System for Computational Discrete Algebra. 20 May 2009 <http://www.gap- system.org/~history/Biographies/Riemann.html>. • "Riemann integral -." Wikipedia, the free encyclopedia. 20 May 2009 <http://en.wikipedia.org/wiki/ Riemann_integral>. • "Riemann Zeta Function -- from Wolfram MathWorld." Wolfram MathWorld: The Web's Most Extensive Mathematics Resource. 20 May 2009 <http://mathworld.wolfram.com/RiemannZetaFunction.html>. • Schechter, Bruce. "143-Year-Old Problem Still Has Mathematicians Guessing - The New York Times." The New York Times - Breaking News, World News & Multimedia. 2 June 2002. 20 May 2009 <http:// www.nytimes.com/2002/07/02/science/143-year-old-problem-still-has-mathematicians- guessing.html?scp=2&sq=Bernhard%20Riemann&st=cse>.