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Number Theory Euler/Riemann Point of Departure in Riemann's 1859 paper containing RH

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Greg

Perseverance
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Feb 5, 2013
1,401
In his 1859 paper entitled "On the Number of Primes Less than a Given Magnitude", Riemann gives as his point of departure the equation

\(\displaystyle \prod\frac{1}{1-\frac{1}{p^s}}=\sum\frac{1}{n^s}\)

where $p$ is all primes and $n$ is all natural numbers. The function of the complex variable $s$, wherever these expressions converge, is called by Riemann $\zeta(s)$.

Any thoughts on how to prove this equation?

All comments welcome. :)
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
9,265
It is called the Euler product formula for the Riemann zeta function.

Wiki gives 2 proofs here.