# Mertens conjecture

Franz Mertens conjectured that $$ where the Mertens function^{} is defined as

$$M(n)=\sum _{i=1}^{n}\mu (i),$$ |

and $\mu $ is the Möbius function^{}.

However, Herman J. J. te Riele and Andrew Odlyzko have proven that there exist counterexamples beyond ${10}^{13}$, but have yet to find one specific counterexample.

The Mertens conjecture^{} is related to the Riemann hypothesis^{}, since

$$M(x)=O({x}^{\frac{1}{2}})$$ |

is another way of stating the Riemann hypothesis.

Given the Dirichlet series of the reciprocal of the Riemann zeta function^{}, we find that

$$\frac{1}{\zeta (s)}=\sum _{n=1}^{\mathrm{\infty}}\frac{\mu (n)}{{n}^{s}}$$ |

is true for $\mathrm{\Re}(s)>1$. Rewriting as Stieltjes integral,

$$\frac{1}{\zeta (s)}={\int}_{0}^{\mathrm{\infty}}{x}^{-s}\mathit{d}M$$ |

suggests this Mellin transform^{}:

$$\frac{1}{s\zeta (s)}=\left\{\mathcal{M}M\right\}(-s)={\int}_{0}^{\mathrm{\infty}}{x}^{-s}M(x)\frac{dx}{x}.$$ |

Then it follows that

$$M(x)=\frac{1}{2\pi i}{\int}_{\sigma -is}^{\sigma +is}\frac{{x}^{s}}{s\zeta (s)}\mathit{d}s$$ |

for $$.

## References

- 1 G. H. Hardy and S. Ramanujan, Twelve Lectures on Subjects Suggested by His Life and Work 3rd ed. New York: Chelsea, p. 64 (1999)
- 2 A. M. Odlyzko and H. J. J. te Riele, “Disproof of the Mertens Conjecture.” J. reine angew. Math. 357, pp. 138 - 160 (1985)

Title | Mertens conjecture |
---|---|

Canonical name | MertensConjecture |

Date of creation | 2013-03-22 16:04:25 |

Last modified on | 2013-03-22 16:04:25 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 10 |

Author | PrimeFan (13766) |

Entry type | Conjecture |

Classification | msc 11A25 |

Synonym | Mertens’ conjecture |

Synonym | Mertens’s conjecture |