PlanetMath: Mertens conjecture

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Mertens conjecture

(Conjecture)

Franz Mertens conjectured that $\left\vert M(n) \right\vert < \sqrt {n}$ where the Mertens function is defined as

$\displaystyle M(n) = \sum_{i = 1}^n \mu(i),$

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

$\displaystyle M(x) = O(x^\frac12)$

is another way of stating the Riemann hypothesis.

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

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

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

$\displaystyle \frac{1}{\zeta(s)} = \int_0^{\infty} x^{-s}dM$

suggests this Mellin transform:

$\displaystyle \frac{1}{s \zeta(s)} = \left\{ \mathcal{M} M \right\}(-s) = \int_0^\infty x^{-s} M(x) \frac{dx}{x}.$

Then it follows that

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

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

Bibliography

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)

"Mertens conjecture" is owned by PrimeFan. [ full author list (3) | owner history (4) ]

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Other names:

Mertens' conjecture, Mertens's conjecture

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Cross-references: Mellin transform, integral, Riemann zeta function, reciprocal, Dirichlet series, Riemann hypothesis, counterexamples, Andrew Odlyzko, Möbius function, Mertens function
There are 3 references to this entry.

This is version 7 of Mertens conjecture, born on 2006-07-09, modified 2007-01-08.
Object id is 8130, canonical name is MertensConjecture.
Accessed 1823 times total.

Classification:

AMS MSC:

11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)

Pending Errata and Addenda

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