12 problems

For an integer $q$, let $\varrho(q)$ denote the number of square roots of $1$ modulo $q$, let $\operatorname{rad}(q)$ be its radical, and let $x_{q;R,NR}$ be the generalized Skewes…

Type-III bias conjecture. As $x\rightarrow\infty$,

Type-II bias conjecture. For any two residue classes $a_i$ and $a_j$,

Let … Here $p$ ranges over primes, and logarithmic scale is measured by logarithmic density, using the variable $y=\log x$; thus a property holds for almost all $x$ in logarithmic…

Negative bias conjecture. There is a bias towards negative values in the distribution of $D_1$: for more than half of the $x\in[2,\infty)$ in logarithmic scale, more than half of t…

Let $L/\mathbb Q$ be a number-field extension and let $L_0/\mathbb Q$ be its Galois closure. Write … where the union is a multiset. LI− conjecture. The multiset…

Double-exponential generalized Skewes-number conjecture. As $\rho(q)/\log q'$ tends to infinity,

Let $r\geq 3$ and let $a_1,\ldots,a_r$ be distinct nonzero integers. Define $\mathcal{Q}_{a_1,\ldots,a_r}$ to be the set of positive integers $q$ such that the $a_i$ are distinct m…

Fix a modulus $q$, and consider the zeros of Dirichlet $L$-functions attached to primitive characters modulo $q$. For a zero $\varrho=\beta+i\gamma$, call $\gamma$ its imaginary pa…

Let $k$ be a modulus, let $\varphi(k)$ denote Euler's totient, and let $a_1,\ldots,a_{\varphi(k)}$ be a permutation of the reduced residue classes modulo $k$. Write $\pi(m,k,a)$ fo…

Rubinstein–Sarnak conjecture. When $r\geq 3$, the race $\{q;a_1,\dots,a_r\}$ is unbiased if and only if $r=3$ and the residue classes $a_1,a_2,a_3$ satisfy assumption (1.3).

Let $A\subset \mathbb{Z}_{>1}$ be a primitive set, meaning that no member of $A$ divides another, and let … Let $\mathcal P$ denote the set of primes. Erdős's primitive set conject…