# Definition:Riemann Zeta Function/Historical Note

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## Historical Note on Riemann Zeta Function

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

In that paper he made several statements about that function.

Studying it led Jacques Salomon Hadamard to his proof of the Prime Number Theorem.

All of the statements made in that paper have now been proved except for one.

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

When Riemann first investigated this function, he used $s$ instead of the more typical complex variable $z$.

To this day, $s = \sigma + i t$ is still typically used as the complex variable in investigations of the zeta function.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.32$: Riemann ($\text {1826}$ – $\text {1866}$)