Bernhard Riemann


("Bernhard riemann," 1868)

Bernhard Riemann was born in 1826 in the Kingdom of Hanover which is Germany today. He was born to Friedrich Bernhard Riemann and Charlotte Ebell. Bernhard has five siblings, one brother and four sisters. Friedrich was a Luthern Pastor as well as a teacher to all six children. He taught Bernhard until the age of ten at which time a local school teacher took over (O'Conner, 1998).

When Bernhard turned fourteen, he entered into a sort of prep school at the Lyceum. At this time, he lived with his grandmother, but unfortunately, she passed away a couple years later. To this, Bernhard moved to the Johanneum Gymnasium. Here he worked hard in the subjects Hebrew and theology, but his main interest was in mathematics. The director of the gymnasium allowed Bernhard to study mathematics from his own personal library (O'Conner, 1998).

In 1846 Bernhard attented the University of Gottingen. Since Riemann was very close to his family, he entered into the theology faculty because his father wanted him to study theology. Because of his interest in mathematics, Riemann wrote to his father and asked for permission to transfer to the faculty of philosophy, which his father generously granted. Once in the mathematics department, Bernhard was able to take math classes, some of which were taught by Moritz Stern and Gauss (O'Conner, 1998) (Rockmore, 2005, pg 66).

Next, Riemann moved to Berlin University to study under many great mathematicians including Steiner, Jacobi, Dirichlet and Eisenstein. During his time at the university, Bernhard studied the use of complex variables and formed the basis for his greatest discovery. In 1849, Bernhard came back to the university and in 1851, submitted his Ph.D. thesis. Again, Riemann worked with many mathematicians, but it was through Weber and Listing that he gained a strong knowledge in theoretical physics and topology (O'Conner, 1998).

For his thesis, Riemann studied complex variables and what we now call Riemann surfaces. Here is where topological methods were introduced into complex funcion theory. Bernhard's thesis was such a great original piece of work that Gauss recommended Riemann to a post at the university working for his Habilitation. This would allow him to become a lecturer. There he worked on his dissertation in which he gave the conditions of a function to have an integral. To complete his Habilitation, Riemann had to give a lecture. In this lecture, Riemann discussed two points, one of which was the definition of curvature tensor. The second was to pose deep questions about geometry and how it relates to the world we live in. Because of the deep nature of the lecture, it was too far above most of the scientists of the time. In fact, it was not completely understood until many decades later (O'Conner, 1998).

Since the lecture was so magnificent, a couple years later, Riemann was appointed as a professor. That same year, Riemann was appointed to a member of the Berlin Academy. This was a great honor for such a yound mathematician. To accept this honor, Riemann submitted a paper titald "On the Number of Prime Numbers Less Than a Given Quantity" (Derbyshire, 2003, Pg. 31). This paper influced mathematics to no end and contained the formula for what is now called the Riemann Hypothesis (Rockmore, 2005, Pg 63).

Riemann married in 1862 to Elise Koch and had a daughter. A few months after being married, Riemann caught a cold which turned into tuberculosis. He had never had good health from the time he was little, so it wasn't much of a surprise that a cold turned very serious for him. For the last couple years of his life, Riemann traveled to Italy where there was warmer weather to help his health but visited Gottingen many times. In 1866, Riemann eventually dies from the tuberculosis (O'Conner, 1998). Riemann now has 79 topics in mathematics named after him.

The Riemann Hypothesis deals with the number of primes that exist under a certain integer. Before going into much detail about his hypothesis, I will explain the questions and background about primes. A huge question still today is how many primes are there less than a given quantity? From the numbers 1 to 1000, there are 168 primes. If one looks at the list of primes, they will see that the number of primes thin out as they go along. For example, z'between 1 and 100 there are 25 primes; between 401 and 500, 17; and between 901 and 1000, only 14" (Derbyshire, 2003, pg33). Continuing the list to one million, one will see that there are only 8 primes in the last hundred block of one million. So, do the primes eventually thin out to nothing? The answer is no. Euclid found that there are always more primes and there is no biggest prime. Since mathematicians proved that there are not a finite number of primes, their next question was whether they could find a law or rule to describe the nature of the primes. Is there a formula that will give the number of primes less than a given number? This brings us back to Riemann's question. After doing some research, Riemann discovered what we call The Prime Number Theorem (Derbyshire, 2003, pg. 33-45). This is pi*(N)~N/logN, pronounced as N over pi of N tends asymptotically to log N. By discovering the Prime Number Theorem, Bernhard came up with what we call the Riemann Hypothesis which states "All non-trivial zeros of a zeta function have real part one-half. Riemann's Hypothesis is still unsolved today (Dudley, 2009, pg. 121-123).

The Music of the Primes--Film

Bringing the Riemann Zeta Function to the World's Attention

Another one of Riemann's greatest contributions was the Riemann Integral or Riemann Sum. A condensed version of the defintion for a Riemann Integral is the area under the curve to the x-axis on a certain interval. The Riemann Sum is an approximation of the Riemann Integral (McLeod, 1980, pg. 8). Riemann took the idea of area under a graph and extended it to graphs of nonlinear functions. The way he did this was to look at the rate of change for very small intervals, treat the function as nearly constant, mulitply rate by time and sum up the resulting displacements. In other words, he was summing up the areas of very narrow rectangles. Although Riemann worked with rectangles of many widths, modern calculus uses rectangles of equal width whose heights are determined by the function value at the left-hand, right-hand, or the mid-point of the rectangle. Modern day calculus has begun to use trapezoids instead of rectangles to estimate a Riemann Integral (Hill, 2011, pg 159).

A useful applet I found online can help demonstrate how using trapezoids over rectangles can be useful. Click Here

The area for negative functions are treated the exact same as for positive functions. If someone asked you what the area of your bedroom was, the answer wouldn't be a negative number. It just doesn't make sense. The same goes for a negative function. Even though a function is negative, it will still have a positive area. "The definite integral is a Riemann sum, so on a given interval where all function values are positive, the sum of the products of function values and positive x-values will always be positive. If a function if negative on a given interval, then the sum of the products of negative function values and positive x-values will produce a negative value" (Hill, 2011, pg 173). If you are wanting area, then you will need to adjust the result.

Below is an applet I created that demonstrates the relationship and closeness of a Riemann Integral and a Riemann Sum of any function you wish.

Left Riemann Sum

Activity: Deriving the Prime Number Theorem

List of Primes

Worksheet

Worksheet Answers

Sources:

Rockmore, Daniel N. (2005). Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers. Westminster, MD: Knopf Publishing Group

Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the greatest unsolved problem in mathematics. Washington, DC: Joseph Henry Press.

O'Conner, J. J. (1998, September). Georg friedrich bernhard riemann. Retrieved from http://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.html

Dudley, Underwood (2009). A Guide to Elementary Number Theory. The United States of America:The Mathematical Association of America. (1868). Bernhard riemann. (1868). [Print Photo]. Retrieved from http://saints.sqpn.com/ncd05166.htm

Hill, Greg (2011). Everything Guide To Calculus I : A Step-By-Step Guide To The Basics Of Calculus-In English! Avon, MA: F+W Media

McLeod, Robert M. (1980). The Generalized Riemann Integral. United States of America: The Mathematical Association of America.