# Riemann Hypothesis

## Contents

## Hypothesis

All the nontrivial zeroes of the analytic continuation of the Riemann zeta function $\zeta$ have a real part equal to $\dfrac 1 2$.

### Critical Strip

Let $s = \sigma + i t$.

The region defined by the equation $0 < \sigma < 1$ is known as the **critical strip**.

### Critical Line

Let $s = \sigma + i t$.

The line defined by the equation $\sigma = \dfrac 1 2$ is known as the **critical line**.

Hence the popular form of the statement of the Riemann Hypothesis:

*All the nontrivial zeroes of the Riemann zeta function lie on the***critical line**.

## Zeroes

Some of the zeroes of Riemann $\zeta$ function are positioned as follows:

### Trivial Zeroes of Riemann Zeta Function are Even Negative Integers

Let $\rho = \sigma + i t$ be a zero of the Riemann zeta function not contained in the critical strip:

- $0 \le \map \Re s \le 1$

Then:

- $s \in \set {-2, -4, -6, \ldots}$

These are called the **trivial zeros** of $\zeta$.

### First zero

The first zero of the Riemann $\zeta$ function is positioned at:

- $\dfrac 1 2 + i \left({14 \cdotp 13472 \, 5 \ldots}\right)$

## Hilbert $23$

This problem is no. $8a$ in the Hilbert $23$.

## Also see

- All Nontrivial Zeroes of Riemann Zeta Function are on Critical Strip
- Critical Line Theorem: an infinite number of nontrivial zeroes exist on the critical line, whatever their multiplicity

## Source of Name

This entry was named for Georg Friedrich Bernhard Riemann.

## Historical Note

The Riemann Hypothesis was stated by Bernhard Riemann in his $1859$ article *Ueber die Anzahl der Primzahlen under einer gegebenen Grösse*.

It is the last remaining statement which has not been resolved is the Riemann Hypothesis.

This problem is the first part of no. $8$ in the Hilbert 23, and also one of the Millennium Problems, the only one to be in both lists.

In Riemann's words, in his posthumous papers:

*[ These theorems ] follow from an expression for the function $\zeta \left({s}\right)$ which I have not simplified enough to publish.*

As Jacques Salomon Hadamard put it:

*We still have not the slightest idea of what that expression should be. ... In general, Riemann's intuition is highly geometrical; but this is not the case for his memoir on prime numbers, the one in which that intuition is the most powerful and mysterious.*

In $1914$, Godfrey Harold Hardy proved the Critical Line Theorem, that there exist an infinite number of nontrivial zeroes of the Riemann $\zeta$ function on the critical line.

This was again demonstrated in $1921$, by Godfrey Harold Hardy together with John Edensor Littlewood.

In $1974$, Norman Levinson demonstrated that At Least One Third of Zeros of Riemann Zeta Function on Critical Line.

By $1983$, systematic exploration of the critical strip with the aid of computers had shown that the first $3 \, 500 \, 000$ nontrivial zeroes were all located on the critical line.

While this is compelling, it is far from being a proof.

In December $1984$, it was announced that Hideya Matsumoto had found a proof, but this was shown to be flawed.

## Sources

- 1983: François Le Lionnais and Jean Brette:
*Les Nombres Remarquables*... (previous) ... (next): $0,5$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $0 \cdotp 5$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.32$: Riemann ($\text {1826}$ – $\text {1866}$) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $0 \cdotp 5$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $14 \cdotp 13472 \, 5 \, \ldots$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Riemann hypothesis** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Riemann hypothesis** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Riemann hypothesis** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Appendix $18$: Millennium Prize problems: $3$