14 problems

Let … where $\mu$ is the Möbius function. Mertens' conjecture. The inequality … should hold for all $x>1$. This conjecture was based on Mertens' hand computation of $M(n)$ through…

Gonek's conjecture. The normalized absolute value of the Mertens function has bounded limit superior:

Define the Mertens function by $M(x)=\sum_{n\leq x}\mu(n)$, where $\mu(n)$ is the Möbius function. Weak Mertens conjecture. … The source states that this conjecture implies the Rie…

Let $M(n)=\sum_{k=1}^n\mu(k)$ denote the Mertens function, where $\mu$ is the Möbius function. Odlyzko–te Riele conjecture. … If true, the sequence $S_n$ discussed in the paper wou…

Let $M(x)=\sum_{n\leq x}\mu(n)$ denote the summatory Möbius function. Ng's large-deviation conjecture. There exists a constant $B>0$ such that … This conjecture concerns the precis…

Unbounded normalized Mertens function conjecture. The quantity $\lvert M(X)/\sqrt X\rvert$ is unbounded. This is a classical conjecture related to the size of Mertens sums; the sou…

Kotnik and van de Lune's conjecture. The smallest counterexample satisfies

Sterneck's conjecture. For every integer $n>200$,

Wei's conjecture. Without treating the values $\mu(n)$ as an independent random sequence, the upper bound for $M(n)$ should have a similar formula to either of these statistical bo…

Mertens square-root upper-bound conjecture. There exist a constant $C>0$, described in the source as very large, and an integer $n_0$ such that for every $n>n_0$,

Large-coefficient square-root upper-bound conjecture. Without assuming that $\mu(n)$ is an independent random sequence, the upper bound for $M(n)$ should have a similar form to the…

Let $k\geq2$, let $M_k(x)$ denote the error in the counting function of $k$-free numbers, and let $C_k>0$ be a constant depending on $k$. Maximum-order conjecture. For every…

Let $\mathcal{M}_n$ be the symmetric matrix introduced in the paper, and let $\Vert \mathcal{M}_n \Vert$ denote its quadratic norm, namely its spectral radius. Cardinal's conjectur…

Let $\mu(k)$ be the Möbius function and let … be the Mertens function. Let $\mathcal{M}=\left(\mathcal{M}_n\right)_{n\in\mathbb{N}^*}$ be the sequence of symmetric square matrices…