| | Ivars Peterson's MathTrek |
June 21, 1999
Mathematicians exhibit such a fascination when they delve into the mysteries of prime numbers.
A prime is a whole number (other than 1) that is evenly divisible only by itself and 1. The list of primes starts off with 2, 3, 5, 7, 11, 13, and 17, and it continues indefinitely. The existence of infinitely many primes, however, gives no indication of how these numbers are distributed among the integers.
Looking at a list of consecutive primes, you can see that, though they appear haphazardly distributed, they gradually get more scarce as the numbers get larger. Nonetheless, no matter how high you go, you are bound to run into a prime. When you get above the enormous number 10400, for example, you find the following primes: 10400 + 69, 10400 + 1449, 10400 + 1987, 10400 + 2877, and 10400 + 2907.
One way to characterize the distribution of primes is to count how many there are up to a given point. There are 4 primes among the first 10 integers; 25 among the first 100; 168 among the first 1,000; 1,229 among the first 10,000; 9,592 among the first 100,000; and 78,498 among the first 1,000,000. As you move from one power of 10 to the next, the ratio of the prime count to the stopping-point number is about 1 to 2.3.
In 1896, mathematicians rigorously proved that, indeed, there is such a trend (see Prime Theorem of the Century, Dec. 23, 1996). The number of primes up to a given number N is approximately N/log N, where log stands for the natural logarithm. The natural logarithm of a number x is 2.302585. . . times the base 10 logarithm of x. The formula predicts that up to 1 billion, for example, the number of primes is about 48 million. The exact count is 50,847,534.
Although the expression N/log N is a good approximation for the prime count, it isn't as accurate as mathematicians would like, and they tried to improve upon it. Those proposed improvements often involve an expression known as the Riemann zeta function.
The zeta function goes back to the Swiss mathematician Leonhard Euler (1707-1783), who expended considerable effort trying to work out rules that govern primes. One of Euler's interests was infinite series: the sums of an infinite number of terms. He proved, for example, that the infinite series 1 + 1/2 + 1/3 + 1/4 + 1/5 + . . . + 1/n + . . . does not have a finite sum.
If the series is modified slightly by raising the denominators of the fractions to a power greater than 1, however, the series has a definite sum. For instance, the infinite series 1 + 1/22 + 1/32 + . . . + 1/n2 . . . has the exact sum 
. Similarly, the series 1 + 1/21.1 + 1/31.1 + . . . 1/n1.1 + . . . adds up to a definite value.
Euler incorporated all series of this type into a single expression now known as the zeta function: zeta(s) = 1 + 1/2s + 1/3s + . . . + 1/ns + . . ., where s can take on any value greater than 1. The examples given above represent two possibilities: zeta(2) and zeta(1.1).
He also discovered a remarkable relationship between the zeta function and prime numbers. The zeta function can be rewritten as an infinite product (an infinite string of terms multiplied together), where each term in the product involves a different prime number. The zeta function, therefore, encodes something about the behavior of primes.
In the 19th century, the German mathematician Georg Friedrich Bernhard Riemann (1826-1866) extended the definition of the zeta function so that, instead of being restricted to real numbers greater than 1 (such as 1.1 and 2), the variable s can be a complex number.
A complex number has two parts and can be written as a + bi, where a is the "real" part and bi is the so-called "imaginary" part, with i representing the square root of -1. Such numbers can be plotted as points on a graph. Each complex number has a "real" x coordinate and an "imaginary" y coordinate. The complex number 3 + 4i would be plotted as the point (3, 4), for example, on what mathematicians term the complex plane.
Riemann was interested in the question of when the zeta function has a zero value. This was crucial for his effort to understand the distribution of primes. Somehow, the zeros of the Riemann zeta function incorporate the deepest secrets of primes, including intimate details of how they are distributed among the integers.
Riemann found that his zeta function is zero for the values -2, -4, -6, and so on. More importantly, he discovered that all other solutions of the equation lie within a thin strip of the complex plane, between the real values 0 and 1. He went on to suggest that those solutions all lie precisely on a straight line in the middle of the strip, represented by complex numbers of the form 1/2 + bi.
That suggestion is now known as the Riemann hypothesis. Many mathematicians consider it to be the most important (and tantalizing) unsolved problem in mathematics.
"If the Riemann hypothesis does turn out to be true and the zeros of the zeta function really are so well ordered, then the connection with the [prime-counting function] will enable even more information about the prime numbers to be deduced than is at present known," says Keith Devlin of St. Mary's College of California in Moraga. "That is what makes it such an important problem for the mathematician at large."
If it were solved, "you would really understand integers," remarks number theorist Peter Sarnak of Princeton University. You would also get drastically improved estimates of the distribution of primes.
Mathematicians have established that there is an infinite number of such solutions of the Riemann equation. The big question is whether all the solutions truly lie exactly on the specified line.
Riemann himself figured out several of those infinitely many solutions--in effect, the values b in the expression 1/2 + bi. These computations may have led him to make his daring conjecture in the first place. In 1903, another mathematician published the values of the first 15 solutions, or zeros, of the Riemann zeta function, providing the first solid evidence supporting the Riemann hypothesis. The first zero occurs when b is approximately 14.134725 and the tenth when b is about 49.773832.
For some mathematicians, the purpose of such computations was to find a counterexample--a point at which the zeta function has the value zero but is not on the line. Modern computers have allowed mathematicians to extend those calculations to the first 1.5 billion zeros, with nary a counterexample in sight.
These results, however, represent just a tiny fraction of all solutions and so may not be typical. That leaves mathematicians still very far from a proof of the Riemann hypothesis.
In recent years, the picture has started to brighten. "There's a sort of euphoria," Sarnak says. "Things are beginning to fall into place."
Mathematicians are starting to exploit a remarkable link between the Riemann zeta function, prime numbers, and quantum mechanics, specifically the spectrum of energy levels exhibited by a quantum system such as an atom.
Those intriguing connections were the subject of a 5-month program just concluded at the Mathematical Sciences Research Institute in Berkeley, Calif. It brought together experts in number theory, quantum mechanics, statistical mechanics, and other areas of physics and mathematics to work on various problems centered on so-called random matrix theory.
Random matrix theory "is a very active area of mathematical research with many applications," Sarnak says. And it could provide insights into how to prove the Riemann hypothesis.
To be continued... .
Copyright 1999 by Ivars Peterson
References:
Cipra, B. 1996. Prime formula weds number theory and quantum physics. Science 274(Dec. 20):2014.
Katz, N.M., and P. Sarnak. 1999. Zeroes of zeta functions and symmetry. Bulletin of the American Mathematical Society 36(January):1.
Peterson, I. 1995. Cavities of chaos. Science News 147(April 29):264.
Richards, I. 1978. Number theory. In Mathematics Today: Twelve Informal Essays. L.A. Steen, ed. New York: Springer-Verlag.
Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.