Mangoldt summatory function

A number theoretic function used in the study of prime numbersMathworldPlanetmath; specifically it was used in the proof of the prime number theoremMathworldPlanetmath.

It is defined thus:


where Λ is the Mangoldt functionMathworldPlanetmath.

The Mangoldt summatory function is valid for all positive real x.

Note that we do not have to worry that the inequality above is ambiguous, because Λ(x) is only non-zero for natural x. So no matter whether we take it to mean r is real, integer or natural, the result is the same because we just get a lot of zeros added to our answer.

The prime number theorem, which states:


where π(x) is the prime counting function, is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the statement that:


We can also define a “smoothing function” for the summatory function, defined as:


and then the prime number theorem is also equivalent to:


which turns out to be easier to work with than the original form.

Title Mangoldt summatory function
Canonical name MangoldtSummatoryFunction
Date of creation 2013-03-22 13:27:16
Last modified on 2013-03-22 13:27:16
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 9
Author mathcam (2727)
Entry type Definition
Classification msc 11A41
Synonym von Mangoldt summatory function
Related topic ChebyshevFunctions