The proof of the Prime Number Theorem uses complex analytic properties of the Riemann zeta function.
150 years before the proof of the Prime Number Theorem,
the special values of the Riemann zeta function were studied by Euler in the 18th century.
The problem solved by Euler is called the Basel Problem.
It is the problem to calculate the sum of the inverses of the squares.
We want to know the value of zeta of 2,
which is equal to 1 plus the inverse of the square of 2 plus the inverse of the square of 3, and so on.
It was first posed by Mengoli in 1644.
The name of the Basel Problem comes from a city in Switzerland.
Basel is Euler's hometown.
In 1734 Euler solved the Basel Problem.
The answer is the square of pi divided by 6.
Here pi is the circumference of a circle with diameter 1.
The Basel Problem shows a mysterious connection between pi and the squares.
Euler's proof was analytic.
He used the infinite product expansion of the sine function.
He also obtained mysterious formulae for other special values of the Riemann zeta function.
He calculated zeta of 2N,
and showed it is a product of pi to the 2N and a mysterious rational number.
For example, zeta of 4, which is the sum of the inverses of the fourth power of positive integers,
is equal to the fourth power of pi divided by 90.
Today, it is known that the rational numbers appearing in Euler's formula of zeta of 2N have deep number theoretic meaning.
These numbers play an important role in Kummer's theory of cyclotomic fields in the 19th century.
By studying these numbers,
Kummer proved Fermat's Last Theorem in some special cases.
Namely, for many prime numbers P called regular primes,
he proved no positive integers X, Y, and Z satisfy the equation
X to the P plus Y to the P is equal to Z to the P.
Although Kummer could not prove Fermat's Last Theorem in full generality,
his results were great achievements.
Kummer's theory was further generalized by Iwasawa in the middle of the 20th century.
Iwasawa theory becomes one of the central topics in modern number theory.
Today, many deep results on prime numbers are obtained by applying methods from Iwasawa theory.
