The complex field, denoted , is the set of all the complex numbers of the form , where and are real numbers. Addition and multiplication over the field are defined as follows:

Over , we define the real function as follows:

For , is defined as an absolutely convergent infinite series that maps values from unto itself, with the exception of where the sum diverges and has a pole1.

where is the Gamma function, a particularly useful analytic continuation of factorials to non-integral points.

side quest

Show that and are indeed equivalent.

Here's some further context about gamma extensions that might be useful.

The zeta zeroes are what make the zeta function particularly interesting. has trivial zeros2 at . Besides these, all the other zeroes, so far, have been found to have real part .

The Riemann hypothesis, which remains unproven, asserts that indeed ALL the non-trivial zeroes have .


Countless theories in fields as far apart as number theory, quantum mechanics, and cryptography assume that the Riemann hypothesis is true.

Globality of Primes

In , Leonhard Euler proved that is equivalent to , which directly relates the zeta function to the prime numbers. For example, this result can be used to construct a direct proof of Euclid's theorem, which posits that there are infinitely many primes, by equating the two equations at ;

where is the harmonic series that diverges to infinity. Thus, the product must also diverge to infinity, which implies an infinite number of terms terms in the product that are greater than .

side quest

Can you think of an alternative proof of Euclid's theorem?

Euclid's Alternate Proof

Suppose there are only finitely many primes. Let be the largest prime. Take . Then is divisible by all primes less than , so must not be divisible by any of those primes (since it would leave a remainder of ).

Therefore, is either prime or divisible by a prime greater than , which contradicts the fact that we picked to be the largest possible such prime.

Locality of Primes

The above equation gives an estimate of the global distribution of primes and them being unbounded. However, a much stronger result which directly relates to the Reimann hypothesis itself estimates the locality of primes.

Gauss3 posited the prime-counting function as follows:

where is the set of all prime numbers less than . This is more commonly referred to as the prime-counting function. That is, it is a step function over that starts at and increases by at each prime number.

Riemann developed a prime-power counting function . Riemann then showed that the zeta zeroes can be used to very accurately approximate the locality of primes. Here's a nice discussion of the consequences.

  1. A pole of a function is a point in the domain of where the value of is undefined.
  2. A zero of a function is a value such that . A trivial zero is a zero that is particularly uninteresting.
  3. Riemann was, in fact, Gauss's student.