In the final part of his 4-part series, Dave Kester explores Bernhard Riemann, whose work laid the foundation for modern technologies like GPS. Learn about the Riemann Hypothesis, one of math's most famous unsolved problems!
This is the final part of a 4-part series where I share how math has changed my career and my life.
In this series, I’m sharing three heroes of mine in this series who have impacted actuarial science, mathematics, and many parts of our lives. I discussed my first two heroes, Leonhard Euler and Carl Gauss, in the previous two blogs.
My third mathematics hero is Bernhard Riemann, who was born in Germany on September 17, 1826, and passed away on July 20, 1866.
Riemann's Impact on Actuarial Science and Geometry
When I studied for the actuarial exams, we used Riemann sums to approximate integrals by using a finite sum. This technique is still valid, though it’s not on the current exam syllabus. But Riemann’s most interesting work is in math theory. In my previous blog, I mentioned how Gauss tutored Riemann, and Riemann extended much of Gauss’s work. Like Gauss, Riemann was born to a poor family, but this didn’t stop him from making history.
Riemann extended Gauss’s work with non-Euclidean geometry. He created a geometry called the Riemann surfaces that built a physical bridge between light, matter, and space-time. These surfaces can have more than three dimensions, officially expanding geometry beyond Euclid. More importantly, Albert Einstein used Riemann’s non-Euclidean geometry in his theories of relativity.
Without Riemann, Gauss, Euler, and other mathematicians laying the groundwork, Einstein couldn’t have achieved his breakthroughs. For instance, GPS technology relies on Einstein’s theories—and, indirectly, on Riemann’s work—to provide accurate location measurements.
Riemann's Hypothesis and Prime Numbers
As mentioned in my previous blog, Gauss identified a formula for calculating the probability of a number being a prime number, but the formula lacked an error term. Riemann identified this error term in his short but historic 1859 paper. He returned to the zeta function, where Euler made history by connecting prime numbers to natural numbers. Euler and others studied the zeta function for real numbers, but Riemann expanded it by considering complex numbers.
Complex numbers have a real part and an imaginary part. Riemann not only extended the zeta function to complex numbers but also used analytic continuation to define the function for values where it was previously undefined, such as when the input values involved negative integers or fractions that were outside the range of earlier definitions. The mystery deepens when the real part of is strictly between 0 and 1. Riemann hypothesized that all nontrivial zeros of the zeta function have a real part equal to 0.5.
This conjecture, known as the Riemann Hypothesis, is one of the most famous unsolved problems in mathematics. A proof would earn you a million-dollar prize. Although progress has been made, no definitive proof exists. However, it has been proven that all nontrivial zeros occur where the real part of is between 0 and 1.
Remarkably, Riemann’s hypothesis connects prime numbers, which are discrete, to the continuous complex plane—a connection no one saw coming.
The Modern Implications of Riemann's Work
No 19th-century mathematician could have predicted the impact prime numbers would have on our lives today. For example, internet security relies on prime numbers, which are connected to the Riemann Hypothesis. Additionally, many mathematical proofs are built upon this hypothesis. Riemann’s 1859 paper laid the foundation for countless layers of mathematical discovery.
If you’re interested in diving deeper, I recommend The Music of the Primes by Marcus du Sautoy. This book describes Riemann’s hypothesis as a great symphony unfolding—a fitting metaphor for the beauty and complexity of his ideas. It’s the same book I mentioned in my first blog that sparked my interest in math history.
You might be surprised to hear mathematicians described as heroes. However, the contributions of Euler, Gauss, and Riemann—and countless others—continue to shape our world. Mathematics builds on past work—no mathematician is an island. Unlike much of science and technology, math is timeless.
Once something is proven true mathematically, it remains true forever. Most mathematicians work quietly, their discoveries rarely making headline news. Yet their work underpins the science, technology, and art we enjoy today.
Observing Mathematics in Everyday Life
Much of mathematics is about uncovering and understanding the patterns that already exist in the universe. While not everyone has time to read math books for fun, we can all take a moment to observe the shapes and patterns around us. Let me leave you with an example: the sine wave.
For example, a sine wave might seem simple, but it has extraordinary properties. By pairing a triangle with a unit circle and applying some math magic, we can create a sine wave. Sine waves are everywhere—in sound, light, electricity, and more. For example, the number of sunlight hours we receive each day follows a sine wave. This predictable pattern reflects the interplay of the Earth and Sun.
Knowing the shape and properties of a sine wave allows us to understand seemingly trivial facts, such as why the sine of a 30-degree angle equals one-half. This realization gives new meaning to the patterns around us.
Mathematics is making music everywhere. Let’s take a moment to enjoy it together!