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# Mertens conjecture

Franz Mertens conjectured that $\left|M(n)\right|<\sqrt{n}$ 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

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

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

$\frac{1}{\zeta(s)}=\int_{0}^{{\infty}}x^{{-s}}dM$ |

suggests this Mellin transform:

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

Then it follows that

$M(x)=\frac{1}{2\pi i}\int_{{\sigma-is}}^{{\sigma+is}}\frac{x^{s}}{s\zeta(s)}ds$ |

for $\frac{1}{2}<\sigma<2$.

# 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)

Synonym:

Mertens' conjecture, Mertens's conjecture

Major Section:

Reference

Type of Math Object:

Conjecture

Parent:

## Mathematics Subject Classification

11A25*no label found*

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## Corrections

typo and spelling by Mathprof ✓

Mark as conjecture by Lando47 ✓

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extra comma by Mathprof ✓

Sum index by rm50 ✓

Mark as conjecture by Lando47 ✓

suggestions by Mathprof ✓

extra comma by Mathprof ✓

Sum index by rm50 ✓