Vipul: Created page with '==Definition== The '''Riemann prime-counting function''' for a positive real number $x$ is the function: $\Pi_0(x) = \frac{1}{2} \left( \sum_{p^k < x} \frac{1}{...'$

2009-05-05T16:23:19Z

Created page with '==Definition== The '''Riemann prime-counting function''' for a positive real number <math>x</math> is the function: <math>\Pi_0(x) = \frac{1}{2} \left( \sum_{p^k < x} \frac{1}{...'

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==Definition==

The '''Riemann prime-counting function''' for a positive real number <math>x</math> is the function:

<math>\Pi_0(x) = \frac{1}{2} \left( \sum_{p^k < x} \frac{1}{k} + \sum_{p^k \le x} \frac{1}{k} \right)</math>.

In other words, it adds the reciprocal of the exponent for every prime power less than <math>x</math>, but if <math>x</math> 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 <math>x</math>.
* [[First Chebyshev function]] adds up the logarithms of all the primes less than or equal to <math>x</math>.
* [[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 <math>x</math>.

Riemann prime-counting function - Revision history

Vipul: Created page with '==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}{...'

Vipul: Created page with '==Definition== The '''Riemann prime-counting function''' for a positive real number $x$ is the function: $\Pi_0(x) = \frac{1}{2} \left( \sum_{p^k < x} \frac{1}{...'$