In the words of mathematician Paul Erdős, “God may not play dice with the universe, but something strange is going on with prime numbers.” All I can say is that Erdős probably wasn’t wrong. Looking back through the previous topics, prime numbers cropped up in the oddest of places: RSA cryptography and  P/NP, specifically. In both cases, prime numbers were used due to the fact that it’s difficult to find primes, and that’s exactly what makes them useful in cryptography. As the size of numbers grows, it grows increasingly hard to determine whether a number is prime, as there are increasing amounts of possible factors, and throughout the entirety of my (brief) mathematics education I’ve learned that there is no pattern to prime numbers. But what if there was? What if there was a way to count the number of primes up to a certain limit? What if there was a way to predict the distribution of all of the prime numbers? According to the Riemann hypothesis, there is a way.

Bernhard Riemann proposed his hypothesis in 1856, and even though it remains without a proof 160 years later, I believe that the problem will be solved at least within my lifetime, potentially in the next decade. 

The easiest way to understand the behavior of prime numbers is to treat them like humans. As individuals, primes are unpredictable and messy, but in large groups they tend to follow a pattern, and just like humans, the larger the group the better the approximations become. That last statement holds true not just because of evidence in number theory, but because of its truth in other fields, like statistics; one of the easiest and most failsafe ways to reduce variation in statistical samples is to simply increase the size of the sample. If one wishes to approximate the number of primes up to a given limit, the approximations become asymptotic to the real value. (Two functions are asymptotic to each other if, as the functions approach infinity, the ratio between the fucntions approaches one.)

This idea is very important in calculus, not only for its direct applications to limits, but in terms of determining the convergence of series. However, while error shrinks relatively in these situations, the absolute error grows without bound.

The quest for an asymptotic formula of primes found its base in complex analysis, or the rigorous formulation of differential, limit, and integral calculus. Even though prime numbers are strictly real numbers, there are ways to transform the arithmetical features of whole numbers to complex numbers, and mathematicians prefer to use the complex number plane because complex functions behave better than real functions. The result of this complex analysis is the Riemann zeta function, an exact formula for the number of primes up to some limit. The zeta function is written in the form of an infinite series, or the sum of an infinite sequence.

In order to solve an infinite series, one has to determine whether the terms, which consist of the sum of all of the previous terms plus that term, approach some limit as n approaches infinity. Most common functions, including sin(x) can be written as infinite sums, and that transformation combined, eventually, with a Fourier transform, created the Riemann  hypothesis from his zeta function, a complex representation of real numbers. However, there is a very large snag: the most important consequences of the zeta function depend upon an unproven statement. How exactly, did Riemann arrive at this incredible function? Multiple other mathematicians laid the groundwork, especially in the form of the Prime Number Theorem.

There exist multiple empirical approximations of the number of primes, the most famous of which are Gauss’s approximation Legendre’s π(x), and Dirichlet’s Li(x). Relatively, the latter two are much better approximations than Gauss’s. Here are the three formulas and how they compare graphically:

These approximations [π(x) and Li(x)] partially compose the Prime Number Theorem (PNT). At its core, PNT is a response to Euclid’s theorem that primes go on forever; another corollary to Euclid’s theorem is the fundamental theorem of arithmetic, or that every positive integer except for one can be represented, in exactly one way, as the product of two primes.

However, this response yields a very interesting pair of equivalent equations. The first is this sequence,

where p runs through all primes, and s is a constant, probably a whole number. However, this equation is equivalent to the infinite series,

which is also called the zeta function. For the first time, there became a concrete relationship between the distribution of whole numbers and the distribution of primes. In order for this series to converge, however, Euclid’s theorem needs to be true, as if there are finite primes, the second version of the equation will equal infinity. The conclusion, then, is that there are, in fact, infinite primes. What did Riemann do with his zeta function that made it so incredible?

Using a Fourier transform, a form of analytic continuation, to extend the zeta function to all complex numbers other than one, and found an exact estimation function [π(x)] for the number of primes up to a given limit. A simple, but equivalent, expression to that function can be written simply as ‘count the number of primes and prime powers up to a chosen limit, and then give larger primes more weight. For example, if our limit = 12, the unweighted count of primes is this:

2, 3, 22 = 4, 5, 7, 23 = 8, 32 = 9, 11

The weighted count adds the natural logarithm of all of the primes, and the result is:

ln(2) + ln(3) + ln(2) + ln(5) + ln(7) + ln(2) + ln(3) + ln(11) = 10.23

When creating a general formula, the weighted count up to some limit x is as follows,

where the sigma is the sum of all numbers p for which zeta(p)is 0, excluding all trivial zeros. In laymen’s terms, π(x) is a fancy way to count primes up to some limit x is actually a simple expression (xp/p) plus a straightforward equation of x. The most important consequence of this expression is that it reduces proving PNT to proving that the weighted count is asymptotic to x.

Why were the zeros of the zeta function so important? The behavior of a function over the complex plane is completely determined by the zeros [f(x) = 0] and poles [f(x) = infinity ], or rather the behavior at those points. In the case of the zeta function, it has only one pole at s = 1, so we center our inferences and knowledge about the entire function at s = 1.

The result of a different analytic continuation on the zeta function yields an infinite series which contains a term which creates either alternating positive and negative values of one. Riemann didn’t stop there; the work that led to the infamous hypothesis deals with an extremely similar, more convenient function xi(x), a function where it remains exceedingly plausible that all zeros of the function are real, which then transfers to the zeta function in the form that all zeros are complex and in the form 12+it, or are at least in the critical strip. This idea is much easier to consider visually; imagine a normal number line, using the real numbers. Now, at one half, create a vertical line, which progresses in a similar fashion but uses the complex number i, or the aqure root of -1. An actual drawing of this idea takes this form,

and it is this combination of ideas that composes the Riemann hypothesis. In simple terms, the statement of the hypothesis is that all nontrivial zeros of the zeta function have real parts on the interval (0,1), as this implies that the sum of all zeros is finite. This, in turn, proves that for large values of x, the number of primes does not matter in the weighted count equation, only x does. Finally, this implication would mean that the zeta function is asymptotic to x. In the simplest form possible, the Riemann hypothesis is that the zeros of the zeta function are not important in the distribution of primes.

In my (and many mathematicians’) opinion(s), the greatest implication of the Riemann hypothesis is that proving it would in turn prove the generalized Riemann hypothesis, which we’ll refer to as GRH. The GRH comes from a refined definition of what it means to be a prime number, as all prime numbers (barring two) are odd. One such way is to classify them as either 4n + 1 or 4n + 3, where n is some variable; other ways to classify include that all primes but 2 and 3 can be written as 6n + 1 or 6n + 5, and all but 5 can be written as either 5n + 1, 5n + 2, 5n + 3, or 5n + 4. How many of each kind of prime are there? In the case of 4n + 1 and 4n + 3, it’s fairly easy to prove that there are an infinite amount of 4n + 3, because each 4n + 3 that isn’t prime has some factor that is prime and of the form 4n + 3. Proving that there are an infinite amount of primes for 4n + 1 is much harder, as this is not the case. It’s so difficult that no one made any progress until Peter Gustav Lejeune Dirichlet in 1837, when he defined analogs of the zeta function, which are known as Dirichlet L-functions. In the 4n + 1 and 4n + 3 case, the L-function is this:

The coefficients are +1 for 4n + 1 and -1 for 4n + 3, and the odd symbol is the Dirichlet character. What matters the most, though, is not the real version of the L-function, but its analytic continuation, which proved that the number of primes of the form 5n + 1x is asymptotic to Li(x)/4, and then proved the same for the other three forms. This led to the conclusion that the zeta function is a special form of the Dirichlet in the form of (k + 0), where zeros of the L-function have the real part 1/2, or the zeros are trivial. The actual statement of GRH is that for every and s where L(s, x) = 0, when 0 < s < 1, then it’s actually 1/2.

Even though there exists no mathematical proof for the Riemann hypothesis, several mathematicians calculated many of the zeros of the zeta function, and they are all in the critical strip. Riemann himself calculated the first few pairs of zeros (as each zero has an equivalent negative counterpart), and in 1903 Jorgen Gram brought the number up to ten pairs. In a cool connection to my second move, Alan Turing discovered a more efficient method to find zeros using a computer in 1953, and he utilized that method to deduce 1104 pairs of zeros; now, however, the current total is 1013 by Yannick Sayer and Patrick Demichel in 2004. However, mathematics has always deplored scientific proof, and even a number like 1013 is relatively small in the grand scheme of things, along with the fact that empirical evidence is not always correct in the world of mathematics.

Even though the Riemann hypothesis remains unsolved, we have reason to believe that it is true, and I personally believe that there will be a proof within my lifetime. And, just like Fermat’s Last Theorem, I believe that the proof will come from someone linking this problem of real analysis to a completely unrelated field of mathematics.