752 problems

Problem. Determine $\kappa$. In particular, is

Let $q$ be fixed, let $a\in\mathbb{F}_q[t]\setminus\{0\}$, and let $M=M(n)$ be in the truncation range specified for the truncated Möbius function. For a monic polynomial $f$, writ…

Let $f:\mathbb{N}\to\mathbb{N}$ be multiplicative. Determine all such functions for which the set of local maxima $\{n\ge 2:f(n)>f(n-1)\text{ and }f(n)>f(n+1)\}$ is finite, and det…

For every integer $n\ge 1$, define $P_n(q)=\prod_{j=1}^{n}(1-q^{3j-2})(1-q^{3j-1})$ and $c_{n,m}=[q^{3m+2}]P_n(q)^4$ for $0\le m\le 4n^2-1$. After all zero coefficients are omitted…

For every integer $n\ge 10$ and every homogeneous cubic form $C\in\mathbb{Z}[x_1,\ldots,x_n]$, there exists a nonzero vector $\mathbf{x}\in\mathbb{Z}^n$ such that $C(\mathbf{x})=0$…

For $n\ge 2$, define $F(n)$ to be the least integer $r\ge 2$ for which there exist integers $1\le a_1<\cdots<a_{r-1}<n$ such that $n!\,a_1!\cdots a_{r-1}!$ is a perfect square. For…

For every integer $k\ge 1$, every collection of distinct shifts $h_1,\ldots,h_k\in\mathbb{Z}$, and every $1$-bounded multiplicative functions…

For each fixed integer $k\ge 2$, define $\alpha_k$ as the supremum of the real numbers $a$ such that, for almost every positive integer $n$, there exist $k$ distinct divisors of…

$\forall n\in\mathbb{Z}$ with $n>2$ and $n\equiv 0\pmod{2}$, there exist primes $p,q$ such that $n=p+q$.

Determine precisely the exponents $0<p<\infty$ for which there exists a constant $C_p<\infty$ such that every Dirichlet polynomial $P$ satisfies…

Determine whether every irrational real number $\alpha$ has infinitely many primes $\ell$ such that $\|\alpha\ell\|<\ell^{-1}$, where $\|x\|$ denotes the distance from $x$ to the n…

For every integer $n\geq 2$ and every cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n$ over a number field, every unramified local component $\pi_v$ is tempered. Equiva…

For every integer $d\ge 2$, every prime $p$, every $H\in[1,p)$ satisfying $\ln\ln p=o(\ln H)$, and every coefficient vector $a=(a_1,\ldots,a_d)\in(\mathbb Z/p\mathbb Z)^d$, one has…

For a family of tempered cuspidal automorphic representations $\pi$ of $\mathrm{GL}_2(\mathbb{Q})$ with bounded archimedean parameters and conductor $q(\pi)$ tending to infinity, d…

Does there exist an absolute constant $C>0$ such that every Dirichlet polynomial $f(s)=\sum_{n=1}^{N}a_n n^{-s}$ satisfies …

Let $r\in\mathbb{Q}_{>0}$ and $k\in\mathbb{Z}_{\ge 1}$. If $\operatorname{Tr}_{\mathbb{Z}}\!\binom{r}{k}_q$ denotes the integer-exponent part (the integer trace) of the Gaussian bi…

For each integer $d\geq 5$, determine the least integer $B(d)$ such that the following holds: for every homogeneous form $F\in\mathbb{Q}[x_1,\ldots,x_n]$ of degree $d$, let…

Let $\lambda(n)=(-1)^{\Omega(n)}$ be the Liouville function and define, for sufficiently large $x$, $f(x):=-\sum_{n\le x}\frac{\lambda(n)\log n}{\sqrt n}\log\frac{x}{n}$. Is…

Determine whether $\lambda^*(L)\ge 0$ for every $L>0$, where $\lambda^*(L)=\inf\left\{Q(f)/\lVert f\rVert_2^2:0\ne f,\ \operatorname{supp}f\subseteq[-L,L]\right\}$ and $Q$ is Weil’…

Let $K$ be a quadratic number field and let $\alpha\in K^\times$ be neither a root of unity nor a square in $K$. Then the set…

Let $\lambda(n)=(-1)^{\Omega(n)}$ be the Liouville function. For every pair of distinct fixed nonnegative integers $h_1,h_2$, one has…

For every integer $k\ge 1$ and every collection of distinct natural numbers $h_1,\ldots,h_k$, where $\lambda(n)=(-1)^{\Omega(n)}$ is the Liouville function, one has…

Let $\omega(n)$ denote the number of distinct prime factors and let $\Omega(n)$ count prime factors with multiplicity. Erdős's conjecture. For $\varepsilon>0$, there are infinitely…

Let $L(s)=\sum a_n n^{-s}$ be an ordinary Dirichlet series with entire continuation of order at most $1$, and let $\mu(\sigma)$ denote its order function on vertical lines. Refined…

Let $\chi\pmod{q}$ be a Dirichlet character, and write $s=\sigma+it$. Generalized Lindelöf hypothesis. For every $\varepsilon>0$, if $\sigma\geq 1/2$ and $|s-1|\geq 1/2$, then … Th…