Prime Number Theorem/Historical Note

Historical Note on Prime Number Theorem
The Prime Number Theorem (PNT) was first conjectured by 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.

Legendre conjectured in $1796$ that there exists a constant $B$ such that $\map \pi n$ satisfies:
 * $\displaystyle \lim_{n \mathop \to \infty} \map \pi n - \frac n {\map \ln n} = B$

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.

was the first one to provide any support for '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 $1891$, and  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 and  in $1896$.

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

The original theorem of used in that proof is given on  as Ingham's Theorem on Convergent Dirichlet Series, which is used in Order of Möbius Function, an essential part of the above proof.

and 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.