What is the Riemann Hypothesis?
By Peace Foo 胡適之
The Riemann hypothesis is well-known enough for the self-described “peasant” Ji-woo to recognize and describe it as “a problem about prime numbers”. To be more precise, it is a question about the Riemann zeta function, which implies other results about the distribution of the primes. It is possibly the most famous unsolved problem in analytic number theory.
Prime numbers, the numbers divisible only by themselves and 1, don’t follow a pattern. But there are results about the number of primes up to a certain number x, denoted π(x). By the 19th century there was a formula Li(x) (which stands for “log integral”) that approximated this value quite accurately, and mathematicians wanted to prove that the ratio Li(x)/π(x) gets as close to 1 as we want [1]. This statement is called the prime number theorem.
Euclid showed that every positive integer is a unique product of primes. By reframing this in terms of real analysis, Euler constructed a function 1 + 2-s + 3-s +4-s + … which was used by other analysts to try to prove the prime number theorem [1]. Riemann extended its definition to include complex numbers s = a + ib. This expanded function is known as the Riemann zeta function. Now he could use some powerful results in complex analysis to attack the problem.
One of those results is that a complex function is completely defined by the values where the function equals 0 or infinity, which are called zeros and poles. Riemann found a pole at s = 1, and zeros at s = -2, -4, -6, … , called “trivial zeros”, that come from the way he extended the function to include complex values. So the important values are the remaining zeros of the zeta function, called “nontrivial zeros”. In 1859, Riemann wrote a paper proposing an exact formula for π(x), which involves Li(x) minus a sum running through the nontrivial zeros. If all these zeros fell between 0 and 1, then this sum would become finite and disappear when compared to large x. That would prove the formula, and the prime number theorem [1].
Not only did every nontrivial zero Riemann find fall between 0 and 1, but they also all had the form 1/2 + it, lying exactly on the line of values of s with real part 1/2, written Re(s) = 1/2. But he couldn’t prove that every single nontrivial zero does this, and this became the famous Riemann hypothesis. The prime number theorem was eventually proved by Hadamard and Poussin independently in 1896 without proving or disproving this very unusual pattern, so the Riemann hypothesis remains unsolved today.
In the movie, Hak-sung shows that all the nontrivial zeros are indeed of this form. There are a few generalizations of the Riemann hypothesis which would have to be proved independently, but once proved there are some important consequences that would follow, several of which concern large prime numbers. In the movie, the National Intelligence Service (NIS) agents are worried about the safety of all the military and commercial encryption methods that rely on such large primes; however, the real threat would come not from a fast primality test, but from a way to efficiently perform prime factorization which is unrelated to the Riemann hypothesis.
The Riemann hypothesis itself also leads to results about the size of gaps between consecutive primes, the precise error in the prime number theorem, and the types of primes that can be found by current methods [1]. Given the board of equations shown at the end of the movie, Hak-sung may be continuing to pursue these or other results in analytic number theory.
References
[1] Stewart, I. (2013). The Great Mathematical Problems. Profile Books.