The Riemann Hypothesis: The Unconquered Everest of Number Theory

The Prime Number The­o­rem gives an ap­prox­i­ma­tion for $\pi(x)$. But how good is this ap­prox­i­ma­tion? The error term in the PNT is deeply con­nect­ed to one of the most famous un­solved prob­lems in all of math­e­mat­ics: the Rie­mann Hy­poth­e­sis.

The Rie­mann Zeta Func­tion $\zeta(s)$

The Rie­mann zeta func­tion, de­not­ed $\zeta(s)$ (pro­nounced "ZAY-tuh of s" or "ZEE-tuh of s"), is a func­tion of a com­plex vari­able $s = \sigma + it$ (where $\sigma$ (sigma) is the real part and $t$ is the imag­i­nary part, and $i = \sqrt{-1}$).

For com­plex num­bers $s$ with real part $\text{Re}(s) = \sigma > 1$, the zeta func­tion is de­fined by the in­fi­nite series:

Leon­hard Euler dis­cov­ered a pro­found con­nec­tion be­tween the zeta func­tion and prime num­bers, known as the Euler prod­uct for­mu­la:

This for­mu­la shows that the zeta func­tion en­cap­su­lates in­for­ma­tion about all prime num­bers.

Bern­hard Rie­mann (1826-1866) ex­tend­ed the de­f­i­n­i­tion of $\zeta(s)$ to almost the entire com­plex plane using a tech­nique called an­a­lyt­ic con­tin­u­a­tion. The an­a­lyt­i­cal­ly con­tin­ued func­tion is still de­not­ed $\zeta(s)$ and is de­fined every­where except for a simple pole (a type of sin­gu­lar­i­ty) at $s=1$.

Zeros of the Zeta Func­tion

A zero of the zeta func­tion is a com­plex number $s$ such that $\zeta(s) = 0$. These zeros are of two types:

1. Triv­ial zeros: These occur at the neg­a­tive even in­te­gers: $s = -2, -4, -6, \ldots$. They are called "triv­ial" be­cause their ex­is­tence is rel­a­tive­ly easy to prove.

2. Non-triv­ial zeros: These are the more mys­te­ri­ous ones. All known non-triv­ial zeros lie on the crit­i­cal line, which is the line in the com­plex plane where the real part of $s$ is $\frac{1}{2}$. That is, $s = \frac{1}{2} + it$ for some real number $t$. These non-triv­ial zeros are typ­i­cal­ly de­not­ed by $\rho = \beta + i\gamma$ (rho equals beta plus i gamma), where $\beta$ (beta) is the real part and $\gamma$ (gamma) is the imag­i­nary part.

The Hy­poth­e­sis

The Rie­mann Hy­poth­e­sis (RH), for­mu­lat­ed by Rie­mann in his ground­break­ing 1859 paper, states:

All non-triv­ial zeros of the Rie­mann zeta func­tion have a real part of $\frac{1}{2}$.

In other words, all non-triv­ial zeros $\rho = \beta + i\gamma$ sat­is­fy $\beta = \frac{1}{2}$. They all lie on the crit­i­cal line $\text{Re}(s) = \frac{1}{2}$.

De­spite enor­mous effort by math­e­mati­cians for over 160 years, the Rie­mann Hy­poth­e­sis re­mains un­proven. It is one of the Clay Math­e­mat­ics In­sti­tute's Mil­len­ni­um Prize Prob­lems, with a $1 mil­lion prize of­fered for its so­lu­tion. Bil­lions of zeros have been com­put­ed, and all of them lie on the crit­i­cal line, pro­vid­ing strong nu­mer­i­cal ev­i­dence, but this does not con­sti­tute a proof.

Sig­nif­i­cance of the Rie­mann Hy­poth­e­sis

Why is this hy­poth­e­sis so im­por­tant? Its truth would have pro­found con­se­quences for the dis­tri­b­u­tion of prime num­bers. Specif­i­cal­ly, it would pro­vide the tight­est pos­si­ble bound on the error term in the Prime Number The­o­rem. If the Rie­mann Hy­poth­e­sis is true, then:

The $O(\sqrt{x} \ln x)$ term (read as "Big O of square root x log x") de­scribes how quick­ly the error $|\pi(x) - \text{Li}(x)|$ grows. The RH im­plies that the primes are dis­trib­uted as reg­u­lar­ly as pos­si­ble, con­sis­tent with their some­what random nature. Prov­ing the Rie­mann Hy­poth­e­sis would in­stant­ly val­i­date hun­dreds, if not thou­sands, of other math­e­mat­i­cal the­o­rems that depend on its truth.