Mertens conjecture

Franz Mertens conjectured that 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

$$\frac{1}{\zeta (s)}=\sum _{n=1}^{\mathrm{\infty }}\frac{\mu (n)}{n^s}$$

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

$$\frac{1}{\zeta (s)}={\int }_0^{\mathrm{\infty }}{x}^{-s}𝑑M$$

suggests this Mellin transform:

$$\frac{1}{s\zeta (s)}=\left\{ℳM\right\}(-s)={\int }_0^{\mathrm{\infty }}{x}^{-s}M(x)\frac{dx}{x}.$$

Then it follows that

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

for .