Prime Number Theorem/Historical Note

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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 $\ds \map \Li x = \int_2^x \frac {\d t} {\map \ln t}$.

It took another century before a proof was found.

Legendre conjectured in $1796$ that there exists a constant $B$ such that $\map \pi n$ satisfies:

$\ds \lim_{n \mathop \to \infty} \map \pi n - \frac n {\map \ln n} = B$

If such a number $B$ exists, then this implies the Prime Number Theorem.

Legendre's guess for $B$ was about $1 \cdotp 08366$, now a historical curiosity known as Legendre's constant.

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 $1$.

In $1891$, Pafnuty Lvovich Chebyshev and James Joseph Sylvester showed that for sufficiently large $x$, there exists at least one prime number $p$ satisfying:

$x < p < \paren {1 + \alpha} x$

where $\alpha = 0 \cdotp 092 \ldots$

Again, since the Prime Number Theorem implies that the above inequality is true for all $\alpha > 0$ for sufficiently large $x$, this constant is also of historical interest only.

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.