The lesser known connection to Prime Numbers

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In his now legendary paper “On the Number of Primes Less Than a Given Quantity”, Riemann stated his famous hypothesis.

While today we know that the Zeta Function is valuable in a wide range of mathematical topics, its first definition was strictly related to the behavior of prime numbers.

In particular, Riemann defined an unusual Prime Counting Function and proved an integral equation that contains his Zeta Function.

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Riemann-von Mangoldt formula
Riemann-von Mangoldt formula
On the number of primes less than a given quantity
On the number of primes less than a given quantity
Weierstrass Factorization Theorem
Weierstrass Factorization Theorem
First Derivative
First Derivative
Analytic continuation for 0
Analytic continuation for 0<Re s<1
Relation to the Gamma Function 1
Relation to the Gamma Function 1

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