Mertens conjecture

Franz Mertens conjectured that |M(n)|<n where the Mertens functionMathworldPlanetmath is defined as


and μ is the Möbius functionMathworldPlanetmath.

However, Herman J. J. te Riele and Andrew Odlyzko have proven that there exist counterexamples beyond 1013, but have yet to find one specific counterexample.

The Mertens conjectureMathworldPlanetmath is related to the Riemann hypothesisMathworldPlanetmath, since


is another way of stating the Riemann hypothesis.

Given the Dirichlet series of the reciprocal of the Riemann zeta functionDlmfDlmfMathworld, we find that


is true for (s)>1. Rewriting as Stieltjes integral,


suggests this Mellin transformDlmfMathworldPlanetmath:


Then it follows that


for 12<σ<2.


  • 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