Riemann Hypothesis

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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