# 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)
Title Mertens conjecture MertensConjecture 2013-03-22 16:04:25 2013-03-22 16:04:25 PrimeFan (13766) PrimeFan (13766) 10 PrimeFan (13766) Conjecture msc 11A25 Mertens’ conjecture Mertens’s conjecture