# Mangoldt summatory function

A number theoretic function used in the study of prime numbers^{}; specifically it was used in the proof of the prime number theorem^{}.

It is defined thus:

$$\psi (x)=\sum _{r\le x}\mathrm{\Lambda}(r)$$ |

where $\mathrm{\Lambda}$ is the Mangoldt function^{}.

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 $\mathrm{\Lambda}(x)$ 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:

$$\pi (x)\sim \frac{x}{\mathrm{ln}(x)}$$ |

where $\pi (x)$ is the prime counting function, is equivalent^{} to the statement that:

$$\psi (x)\sim x$$ |

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

$${\psi}_{1}(x)={\int}_{0}^{x}\psi (t)\mathit{d}t$$ |

and then the prime number theorem is also equivalent to:

$${\psi}_{1}(x)\sim \frac{1}{2}{x}^{2}$$ |

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 |