50 problems

Gonek's conjecture. The normalized absolute value of the Mertens function has bounded limit superior:

Harper's conjecture. For every $\varepsilon>0$, almost surely

Boyd's valuation-four conjecture. The inequality $\nu_p(H_n)\geq 4$ never occurs.

Let $N$ tend to infinity, and let $H$ be the length of the interval in the prime-counting function. The distribution of the normalized values of $\psi(n+H)-\psi(n)$ for $n\in[1,N]$…

Cohen and Martinet's conjecture. For every “reasonable” non-negative function $f$ on the isomorphism classes of finite $eZ_S[Gamma]$-modules,

Maximal-decay conjecture. Despite the irregular behavior of $c_i$, one could tentatively conjecture that $c_i$ decays at the maximal rate $1/i$ consistent with this lower bound. Th…

Let $p_n$ be the $n$th prime and let $\gamma$ be the Euler–Mascheroni constant. Granville's maximal-gap conjecture. The maximal prime-gap limsup satisfies … Granville argued that C…

Let $p_n$ denote the $n$th prime. For $varepsilon>0$ and $0\leqslant \lambda\leqslant (\log x)^{1-varepsilon}$, consider the number of primes $p_n\leqslant x$ whose following prime…

Let $p_n$ denote the $n$th prime, and let $\varepsilon>0$. Poisson Tail Conjecture. For $1\leq H\leq(\log X)^{2-\varepsilon}$, the two gap-counting quantities satisfy … … For…

Let $K$ be a totally real number field of degree $n$. For an odd prime $p$ unramified in $K/\mathbb{Q}$ and a place $\mathfrak{p}$ above $p$, let $\tau^{\mathfrak p}$ be the map ap…

Let $N$ be large and let $\epsilon>0$ be fixed. Suppose that … Define the von Mangoldt function by … and set … Let $\mu_k$ denote the $k$th moment of the standard normal distributi…

Let $n$ tend to infinity, let $\lambda\in\mathbb{R}$, and let $\omega(m)=\sum_{p\mid m}1$ denote the number of distinct prime divisors of $m$. Here $p$ denotes a prime and $\phi$ i…

Let $C_{\Omega}(n)$ be the unsigned arithmetic function used in the source, let $A_0$ and $B_0$ be the absolute constants appearing there, and define … where … Let $\Phi$ denote th…

LeVeque's conjecture. The optimal rate of convergence of $Z_n$ to $N$ in Kolmogorov distance is of order

Let $f$ be a modular form of weight $3/2$ attached to an elliptic curve $E$, with normalized Fourier coefficients $b(n)$, and let $\kappa^\pm$ be a positive constant. For…

Let $X\geq 1$ and let $x$ be drawn uniformly at random from $[X,2X]$. For a positive scale $H$, define the fluctuation process … Here $\mu^2(n)$ is the squarefree indicator. Fracti…

Consider central values of families of $L$-functions, such as quadratic twists $L(f,\chi_D,k)$ of a Hecke eigenform. The conjecture concerns the distribution of their central value…

Let $K_0$ be the fixed base field, let $Gamma$ be the fixed finite group, and let $e_1,dots,e_m$ be the irreducible central idempotents of $Q[Gamma]$. Let $S$ be a finite set of pr…

Let $S_{N,m}:=\sum_{n\in E_{N,m}}z(n)$, where $E_{N,m}:=\{1\le n:\Omega(n)=m\}$ and $z$ is the completely multiplicative Steinhaus random function. Write…

In the setting of Steinhaus chaos, let $c(k)$ denote the constant in the conjectured comparison $\|S_{N,m}\|_{2k}\asymp\|S_{N,m}\|_2$ for $m<\beta\log\log N$ and $\beta<c(k)$. Sixt…

Let $(z(p))_{p\ \text{prime}}$ be Steinhaus random variables, extended completely multiplicatively to $\boldsymbol{N}$. For $E_{N,m}:=\{1\le n:\Omega(n)=m\}$, set…

Let $f:\mathbb{N}\to\mathbb{N}_0$ satisfy $0\leqslant f(n)\leqslant n$, let $\Omega_f$ be the associated random space of numerator sets, and let $W^P(\Psi)$ denote the set of numbe…

For primes $p\ne\ell$, let $D_p=\dim_{\mathbb{Q}}S_2(\Gamma_0(p);\mathbb{Q})$, let $D_p^\pm=\dim_{\mathbb{Q}}S_2(\Gamma_0(p);\mathbb{Q})^\pm$, and define … where the tilded polynom…

Ford–Lamzouri's conjecture. If $2\leq n\leq(\log q)^{1-\varepsilon}$, then uniformly for all $(a_1,\dots,a_n)\in\mathcal{A}_n(q)$,

Let $C(n)$ range over the Apéry-like sequences listed in Tables 2 and 3 of the source, including $(\eta)$ and the sequences $(\delta)$, $(\alpha)$, $(\epsilon)$, $(\zeta)$, and…