# 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 Line

The line defined by the equation $z = \dfrac 1 2 + i y$ 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:

### First zero

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

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

This sequence is A058303 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Hilbert $23$

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

## Also see

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

## Sources

- 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 ($1826$ – $1866$) - 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): $14 \cdotp 13472 \, 5 \, \ldots$ - 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**