Riemann prime-counting function

Definition

The Riemann prime-counting function for a positive real number $x$ is the function:

$\Pi _{0}(x)={\frac {1}{2}}\left(\sum _{p^{k}<x}{\frac {1}{k}}+\sum _{p^{k}\leq x}{\frac {1}{k}}\right)$.

In other words, it adds the reciprocal of the exponent for every prime power less than $x$, but if $x$ itself is a prime power, it adds only half of the reciprocal of the exponent.

Relation with other functions

• Prime-counting function simply counts the number of primes up to $x$.
• First Chebyshev function adds up the logarithms of all the primes less than or equal to $x$.
• Second Chebyshev function is the summatory function for the von Mangoldt function: it adds up the logarithms of all the maximal prime powers less than or equal to $x$.