70 problems

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,

Let $p$ be an odd prime, let $\left(\frac{\cdot}{p}\right)$ denote the Legendre symbol modulo $p$, and define the Fekete polynomial … Montgomery's bounds concern the maximum of…

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 $N$.

Let $n\geq 3$, let $\epsilon>0$, let $M$ be an integer, let $Q\geq 3$ be an integer, let $N\leq Q^{1/n}$ be an integer, and let $\chi$ be a real non-principal character of modulus…

Large-sieve bound conjecture. One has

For $B>0$, define … where $D$ ranges over fundamental discriminants, and let $\rho$ denote the Dickman-de Bruijn function. Dickman-de Bruijn conjecture. One has … for all $B>0$. Th…

For an arithmetic function $f$, define … Let $\chi_0$ denote the principal character modulo $q$. Smooth-number approximation conjecture. There exists a constant $A>0$ such that for…

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…

Gow–McGuire Conjecture 3. For every $\alpha\in\mathbb F_{q^2}\setminus\mathbb F_q$ and every $\mu\in\mathbb F_q^\times$, there exists $\lambda\in\mathbb F_q$ for which…

Gow–McGuire Conjecture 2. For every such irreducible quadratic and every $\mu\in\mathbb F_q^\times$, there exists $\lambda\in\mathbb F_q$ for which $G_{\mu,\lambda}(X)$ is primitiv…

Gow–McGuire Conjecture 1. For every irreducible quadratic $X^2+aX+b\in\mathbb F_q[X]$, there exists $\lambda\in\mathbb F_q$ for which $G_{1,\lambda}(X)$ is primitive over…

Centered subcritical Vaughan moment conjecture. One has

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

Let $p$ be prime, let $\chi$ be a character modulo $p$, and let $\Phi_\chi(V)$ denote the tail distribution considered in the paper. For fixed $V$ in the range of the relevant theo…

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

Harper's sharpness conjecture. The bounds in Theorem should be sharp provided that $x\leq 0.99r$.

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

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…

Fixed-rank GAP energy conjecture. The rank-$2$ estimate

Square-root bound conjecture. For every $\varepsilon>0$,

Let $x,r\to+\infty$, and let $\chi$ vary uniformly over the family of Dirichlet characters modulo $r$. Write $e(t)=\exp(2\pi i t)$ and let $\mathcal{CN}(0,1)$ denote the standard c…

Character-sum cancellation conjecture. For every $\varepsilon>0$ and every $x>q^{\varepsilon}$,

Let $\chi$ be a non-principal character modulo $q$, and let $x$ be a positive real number. Character-sum upper-bound conjecture. For some fixed $\epsilon>0$, … The text states 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.…