Given Mertens Function defined by

(1) |

(2) |

(3) |

Mertens conjecture was proved false by Odlyzko and te Riele (1985). Their proof is indirect and does not produce a specific
counterexample, but it does show that

(4) |

(5) |

It is still not known if

(6) |

**References**

Anderson, R. J. ``On the Mertens Conjecture for Cusp Forms.'' *Mathematika* **26**, 236-249, 1979.

Anderson, R. J. ``Corrigendum: `On the Mertens Conjecture for Cusp Forms.''' *Mathematika* **27**, 261, 1980.

Devlin, K. ``The Mertens Conjecture.'' *Irish Math. Soc. Bull.* **17**, 29-43, 1986.

Grupp, F. ``On the Mertens Conjecture for Cusp Forms.'' *Mathematika* **29**, 213-226, 1982.

Jurkat, W. and Peyerimhoff, A. ``A Constructive Approach to Kronecker Approximation and Its Application
to the Mertens Conjecture.'' *J. reine angew. Math.* **286/287**, 322-340, 1976.

Mertens, F. ``Über eine zahlentheoretische Funktion.'' *Sitzungsber. Akad. Wiss. Wien IIa* **106**,
761-830, 1897.

Odlyzko, A. M. and te Riele, H. J. J. ``Disproof of the Mertens Conjecture.'' *J. reine angew. Math.*
**357**, 138-160, 1985.

Pintz, J. ``An Effective Disproof of the Mertens Conjecture.'' *Astérique* **147-148**, 325-333 and 346, 1987.

te Riele, H. J. J. ``Some Historical and Other Notes About the Mertens Conjecture and Its Recent Disproof.''
*Nieuw Arch. Wisk.* **3**, 237-243, 1985.

© 1996-9

1999-05-26