27 problems

For a positive integer modulus $\ell$ and reduced residue classes $\alpha$ and $\beta$ modulo $\ell$, let $\pi(x,\alpha;\ell)$ count primes $p\leq x$ with $p\equiv\alpha\pmod\ell$.…

Feuerverger–Martin's conjecture. There should exist a bias factor $F_n(e_1,\dots,e_r)$, defined as a linear combination of the quantities $G_n(e_j)$, such that, for ordered $r$-tup…

Let $k$ denote the number of prime factors counted with multiplicity in a product of primes, and consider the bias in the distribution of such products between reduced residue clas…

Let $r\geq 3$, let $k$ be a modulus, and let $a_1,\ldots,a_r$ be reduced residue classes modulo $k$. Write $\{k;a_1,\ldots,a_r\}$ for the corresponding prime-number race, and let…

Feuerverger--Martin conjecture. In this regime,

A race game is associated with an ordered tuple of reduced residue classes modulo a modulus, and an isomorphism is a bijection between tuples that preserves the corresponding race-…

Let $q$ tend to infinity, and let $r_0=r_0(q)$ be any function tending to infinity with $q$. For each modulus $q$, let $delta_{q;a_1,dots,a_{r_0}}$ denote the density associated wi…

Ford's conjecture. Such solutions occur arbitrarily far out when $r=3$, but do not occur arbitrarily far out when $r\geq 4$.

Nonvanishing conjecture. If ${\rm Gal}(L/\mathbb Q)$ admits no irreducible symplectic representations, then the associated Artin $L$-functions do not vanish at $s=1/2$.

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…

Quantitative linear independence conjecture. There exists a constant $c_k>k$ such that, for every $\boldsymbol{\epsilon}\in\{\pm1\}^k$,

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…

Prime number race mod $4$ conjecture.

Conjecture that all primes are Erdős-strong. Every prime $p$ is Erdős-strong.

Let $q$ be sufficiently large, let $\boldsymbol{a}=(a_1,dots,a_n)$ range over $\mathcal{A}_n(q)$, and let $delta(q;a_1,dots,a_n)$ denote the logarithmic density of the ordering…

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

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

Maximal prime-race density conjecture. One might conjecture that the greatest density attainable by any prime number race is

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…