59 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…

Let $x\to\fty$. Let $a_0,a_1,\ldots,a_{k-1}$ be a fixed admissible $k$-tuple, and let $b_0<b_1<\cdots<b_{l-1}$ be a fixed subset of small integers. Hardy–Littlewood–Chowla conjectu…

Gonek–Ng conjecture. The correct order of magnitude of $M(x)$ should be around

Let $(X,T)$ be a minimal topological dynamical system with $X$ a compact metrizable space and $T:X\to X$ a homeomorphism, and suppose that $(X,T)$ has zero topological entropy. Pol…

Upper-bound conjecture. The values $d_n$ are bounded from above by $0.6040$.

Logarithmically averaged Sarnak conjecture. One has

Logarithmically averaged Chowla conjecture. These correlations tend to zero as $N\to\infty$:

Let $\mu(n)$ denote the Möbius function. For $X\to\infty$, let $H=X^\delta$ with $\delta\in(0,1)$ and average over $x\in[X,2X]$. Good-Churchhouse conjecture. … This conjectures tha…

Let $s\geqslant 1$. An $s$-step nilmanifold is a quotient $G/\Gamma=(G/\Gamma,d_{G/\Gamma})$ with smooth metric, and an $s$-step nilsequence is a sequence of the form…

Möbius and Nilsequences Conjecture. $\operatorname{MN}(s)$ holds for every $s\geqslant 1$; equivalently, the Main Theorem for $2$-step nilmanifolds has an analogue for $s$-step nil…

Let $G/\Gamma$ be a $k$-step nilmanifold, let $T_g(x\Gamma)=xg\Gamma$, and let $(F(T_g^n\cdot x))_{n\in\mathbb{N}}$ be a $k$-step nilsequence. The Möbius orthogonality conjecture f…

Let $\mathbb K=\{k_i\}_{i\in\mathbb N}$ be the set of integers produced by the paper's iterative construction, and let … be the set of square-free, or Möbius, numbers. Auil's enume…

Assume the Riemann Hypothesis, so that every nontrivial zero of the Riemann zeta function has the form $\frac12+i\gamma$. Consider the set of positive imaginary ordinates $\gamma$…

Let $S=(q_1,\dots,q_t)$ be a finite sequence of primes and define $\nu_S(n)=(\nu_{q_1}(n),\dots,\nu_{q_t}(n))$. Say that $n$ is $S$-squarefree if $q^2\nmid n$ for every prime…

Let $0<\theta<1$ be fixed, let $N$ be a sufficiently large natural number, and let $g$ be either $g(n)=\mu(n)$ or $g(n)=\mu(n)\log(n)$. Let $\Lambda(n)$ and $\varphi(q)$ denote the…

Let $0<\theta<1$ be fixed, let $N$ be a sufficiently large natural number, and let $\Lambda(n)$, $\mu(n)$ and $\varphi(q)$ denote the von Mangoldt function, Möbius function and Eul…

Let $0<\theta<1$ be fixed, let $h\neq0$ be a fixed even integer, and let $\Lambda(n)$, $\mu(n)$ and $\varphi(q)$ denote the von Mangoldt function, Möbius function and Euler totient…

Let $0<\theta<1$ be fixed, and let $h\neq 0$ be a fixed integer. For a primitive non-principal Dirichlet character and the Möbius function $\mu$, consider the distribution of the t…

Harper's conjecture. For all $0\leq q\leq 1$ and any fixed $A>0$, for large prime $r$,

Let $f(n)$ be a deterministic sequence, let $P(x)$ be a polynomial with non-negative integer values, and say that a sequence $g(n)$ is Möbius disjoint when … where $\mu$ is the Möb…

Let $wp:Z\toZ$ be a bracket polynomial of complexity $delta$, and let $C>0$, $epsilon>0$. Theorem gives bounds in short intervals when $wp$ is an ordinary polynomial, with implied…

Let $wp:Z\toZ$ be a bracket polynomial of complexity $delta$. Theorem gives bounds for the relevant Möbius sums when $wp$ is linear. The bracket-polynomial extension conjecture. Bo…

Let $cmu$ denote the Möbius function, and let $f e c1g^2$ be a polynomial in $cZ[x]$, where $c1 e 0$ belongs to $cQ$ and $g e 0$ belongs to $cZ[x]$. Chowla's conjecture. One has ……

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

Let $\mathcal T(i)=\frac12 i(i+1)$ be the $i$-th triangular number, let $i\leq_{\mathcal T}j$ mean that $\mathcal T(i)$ divides $\mathcal T(j)$, and let $\mu_{\mathcal T}$ be the M…