# Prime Number Theorem/Historical Note

## Historical Note on Prime Number Theorem

The Prime Number Theorem (PNT) was first conjectured by Carl Friedrich Gauss when he was $14$ or $15$, but he was never able to prove it.

He also posited the suggestion that it could be approximated by the Eulerian logarithmic integral $\displaystyle \map \Li x = \int_2^x \frac {\d t} {\map \ln t}$.

It took another century before a proof was found.

Pafnuty Lvovich Chebyshev was the first one to provide any support for Gauss's conjecture when he proved in $1850$ that:

- $\dfrac 7 8 < \dfrac {\map \pi x} {x / \ln x} < \dfrac 9 8$

for all sufficiently large $x$.

He also proved that if the limit of the expression in question *does* exist, then its value must be $1$.

Since then, several complete proofs have been discovered.

The first proofs were given independently by Jacques Salomon Hadamard and Charles de la Vallée Poussin in $1896$.

They relied on the theory of functions of a complex variable.

The original theorem of Hadamard used in that proof is given on $\mathsf{Pr} \infty \mathsf{fWiki}$ as **Ingham's Theorem on Convergent Dirichlet Series**, which is used in Order of Möbius Function, an essential part of the above proof.

Atle Selberg and Paul Erdős would later give an elementary proof of the PNT, in $1948$.

Their proof did not make use of any analytic function theory, and relied entirely on basic properties of logarithms.

Dispute over whether to publish their results jointly or separately created a life-long feud between the two mathematicians.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $-1$ and $i$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.25$: Gauss ($1777$ – $1855$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.31$: Chebyshev ($1821$ – $1894$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.32$: Riemann ($1826$ – $1866$): Footnote $6$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.16$: The Sequence of Primes - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $-1$ and $i$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**prime number theorem**