The Theorem of the Day: The Riemann Explicit Formula

The number of primes not exceeding a given real number $x$ is given by 

$π(x) = R(x) + \sum R(x^ρ)$

where the sum is over all zeros $ρ$ of $ζ$, the Riemann zeta function, and $R(x)$ is the entire function of $log x$ defined by 

$R(x)=1+ \sum_{n-1}^ \infty  \dfrac{(logx)^n}{nn!ζ(n+1)}$.

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