122 problems

Let $\mathcal{M}$ denote the class of multiplicative functions under consideration, and let $f_1,\ldots,f_k\in\mathcal{M}$. Suppose that for some $1\leq j_0\leq k$, the function…

Erdős's conjecture.

Erdős–Wintner conjecture. If $f$ takes only the values $\pm1$, then the limit

Let $a$ be a multiplicative automatic sequence, and let $g$ be a function on the positive integers. A function is eventually periodic if it agrees with a periodic function for all…

Heath-Brown's conjecture. For every completely multiplicative function $f:\mathbb{N}\to[-1,1]$, there is some constant $\delta_1>-1$ for which the displayed lower bound holds. Heat…

Let $\mathcal{M}$ be the class of multiplicative functions, and let $f\in\mathcal{M}$ be real-valued. A Furstenberg system of $f$ is a measure-preserving dynamical system arising a…

Hilberdink–Luca–Tóth conjecture. Asymptotic formulas with error terms exist for $S_n(x)$, $U_n(x)$, and $V_n(x)$ for every $n\geq 3$.

Sun's conjecture. For all $x\geq 3078$,

Goldmakher's conjecture. Let $f\in\mathcal{F}$. Then for any $1\leq y\leq x$,

Harper's conjecture. If $f(n)$ is not identically equal to $1$, then the minimum $\min\{N,q/N\}$ in this inequality can be replaced by $N$.

Kátai's conjecture. If $|f(n+1)-f(n)|\to 0$ as $n\to\infty$, then $f(n)=n^{it}$ for some $t\in\mathbb{R}$.

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

Let $b(n)$ be the indicator of the positive integers representable as sums of two squares, and let $q$ be a fixed positive integer. Connors–Keating conjecture. There is a precise p…

Maximal-projection conjecture. One has

Conjectured optimal bound. For $1/\sqrt e\leqslant \alpha\leqslant 1$, one has

For integers $k\geq 2$ and $q\geq 1$, define the error term $E_{k,q}(x)$ by … where … Here $\varphi(q)$ is Euler's totient function, $\theta(q)$ is the number of positive squarefre…

Let $f:\mathbb N\to\mathbb U$ be a multiplicative function, let $\alpha\in[0,1)$ be approximated by a reduced fraction $a/q$ with $q\leqslant x$ and $|\alpha-a/q|\leqslant 1/(qx)$,…

Let $f:\mathbb N\to\mathbb U$ be a multiplicative function, let $\alpha\in[0,1)$ be approximated by a reduced fraction $a/q$ with $q\leqslant x$ and $|\alpha-a/q|\leqslant 1/(qx)$,…

Let $f:\mathbb N\to\mathbb U$ be a multiplicative function, let $\alpha\in[0,1)$ be approximated by a reduced fraction $a/q$ with $q\leqslant x$ and $|\alpha-a/q|\leqslant 1/(qx)$,…

Let $f:\mathbb N\to\mathbb U$ be a multiplicative function, let $\alpha\in[0,1)$ be approximated by a reduced fraction $a/q$ with $q\leqslant x$ and $|\alpha-a/q|\leqslant 1/(qx)$,…

Let $f:bb N\to\mathbb U$ be a multiplicative function, let $\alpha\in[0,1)$ be approximated by a reduced fraction $a/q$ with $q\leqslant x$ and $|\alpha-a/q|\leqslant 1/(qx)$, and…

Let $E_{2k}(q)$ be the Eisenstein series of weight $2k$, and let $g(q)=a_0+a_1q+a_2q^2+\cdots$ be a power series such that $n\mapsto a_n/a_1$ is a multiplicative function for posit…

Let $x$ be sufficiently large. Consider a real-valued completely multiplicative function $f$ satisfying $|f(n)|\leq 1$ for every positive integer $n$, and suppose that $f(p)=0$ for…

Polynomial exponential-sum conjecture. If $g$ and $h$ satisfy

Harper's conjecture. When $N$ exceeds the scale $\exp(\sqrt{\log q})$, the influence of $f(\cdot)$ should surpass that of $\chi(\cdot)$, potentially leading to a new upper bound fo…