30 problems

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

Let $P\geq 1$, let $k\leq P$ be a nonsquare, so that the character $\left(\frac{k}{\cdot}\right)$ is nonprincipal, and set $H=P^{2/3+\delta}$. Let $e(x)=\exp(2\pi i x)$ and let…

Centered subcritical Vaughan moment conjecture. One has

Prefix sum recurrence conjecture. For an absolute constant $A>0$, independently of $a\in\mathbb{F}_p$,

Erdős's eventual-time conjecture. There exists a constant $C_\varepsilon>0$ such that

Conjecture $C_n$. For every $\varepsilon>0$, there exists $\delta=\delta(n)>0$ such that, for all $M$,

Jutila's conjecture. For all $k=1,2,\dots$ and $X\geq 3$, $Y\geq 1$, there are coefficients $c_1(k)$ and $c_2(k)$ depending only on $k$ such that

Let $U(N)$ be the unitary group, let $\nu_N$ be its Haar probability measure, and let $\operatorname{Sym}^k\mathbb C^N$ be the $k$th symmetric power of the standard representation.…

Let $c\in\{\mu,\lambda\}$, where $\mu$ is the Möbius function and $\lambda$ is the Liouville function. Let $r$ be a prime modulus, let $\mathbb E_\chi$ denote the average over Diri…

Let $q$ be a prime with $q\equiv 3 \pmod 8$, let $x$ be rational with $0<x<1/2$, and define … where $h(q)$ is the class number and $\chi_q$ is the relevant quadratic character. Num…

Let $q$ be squarefree with $q\equiv 3 \pmod 8$, and let $f_q(x)$ denote the function defined earlier in the paper. Positivity conjecture. One has … for $0\leqslant x\leqslant 1/2$.…

Farmer–Gonek–Hughes conjecture. As $q\to\infty$, there is a constant $B$ such that

Let $D$ be a fundamental discriminant, and let $N_D$ denote the number of zeros of the Fekete polynomial $F_D$ in $(0,1)$. Baker–Montgomery conjecture. For almost all fundamental d…

Let $\epsilon>0$ be a real number. For subsets $A,B emathbb{F}p$?

Granville–Soundararajan conjecture. There exists a constant $A>0$ such that, for every non-principal character $\chi$ modulo $q$ and every $1\leq x\leq q$, uniformly,

Infinitude conjecture. There are infinitely many row-dominant characters.

Mean-value conjecture. As $p$ tends to infinity,

Let $p$ be a prime for which the quartic residues are defined, let $F_p$ be the induced subgraph of the Paley graph $G_p$ on the nonzero fourth powers modulo $p$, and let $A_{F_p}$…

Necklace character-sum conjecture. For every $k\geq1$,

General Burgess-like character-sum conjecture. For a certain function $c:\mathbb{N}\rightarrow\mathbb{N}$, one has

Burgess-like character-sum conjecture. For a fixed $\epsilon>0$, one has

Mixed conjecture. There exists a constant $A>0$ such that, uniformly in these ranges,

Let $q$ be large and let $\chi$ be a non-principal character modulo $q$. For real numbers $x$ and $y$ satisfying … and writing $P(n)$ for the largest prime factor of $n$ and…

Let $\eta$ be a real number satisfying $0<\eta\leq 1$. A real character is a Dirichlet character taking real values, and a character is non-principal when it is not the principal c…

Let $p$ be an odd prime, let $0\leq\alpha\leq\frac{1}{2}$, and define … where $\left(\frac{n}{p}\right)$ is the Legendre symbol. Majority nonnegativity conjecture. For every…