# Riemann Hypothesis/Historical Note

## Historical Note on Riemann Hypothesis

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 $\map \zeta s$ 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): $0 \cdotp 5$