The Mertens function is defined as: $$M(n)=\sum_{k=1}^n\mu(n)$$ where $\mu(n)$ is the Möbius function. The Mertens conjecture states: $|M(n)|\lt\sqrt n$. In 1985 A. M. Odlyzko e H. J. J. te Riele proved the conjecture to be false. There exists a number $m$ with $10^{12}\lt m\lt 10^{65}$ for which the previous inequality doesn't occur.
My question is: the number $m$ is the only number for which this happens or is there an infinite (or finite) set of numbers for which the Mertens conjecture is false?
There are infinitely many counter-examples. This was proved by Tadej Kotnik and Herman te Riele in 2006, in their article The Mertens Conjecture Revisited In: Hess, F., Pauli, S., Pohst, M. (eds) Algorithmic Number Theory. ANTS 2006. Lecture Notes in Computer Science, vol 4076. Springer, Berlin