7 problems

Artin's automorphy conjecture. There exists a newform $f\in S(N,\chi,1)$ such that

Let $a>1$ be a fixed integer, and let $(w_n)$ be the sequence defined by … For the generalized Artin problem, take $f(n)=1/n$. Laxton's conjecture. The resulting generalized Artin…

Let $E_{d,g}(x)$ be the error contribution defined by … where $S_{k,l}(x;d,g)$ is the sum defined in the source, and let $[d,g]$ be Artin admissible. Artin prime-pair error-term co…

Fix an even integer $d$ and an integer $g\in\mathbb{Z}\setminus\{-1\}$ satisfying the hypotheses in the source, and let $[d,g]$ be the Artin-admissibility condition. Let $\pi_d(x)$…

Given coprime positive integers $a$ and $d$, let ${\mathcal P}_G(d,a)$ be the set of G-irregular primes congruent to $a$ modulo $d$, and let $\pi(x;d,a)$ count primes $p\le x$ with…

Let ${\mathcal P}_G$ be the set of G-irregular primes, and write ${\mathcal P}_G(x)$ for the number of its elements not exceeding $x$. Let $A$ be the Artin constant, … and let…

Let $F$ be a number field, let $\rho$ be an irreducible $n$-dimensional Artin representation of $\Gamma_F=\operatorname{Gal}(\bar F/F)$, and let $\pi$ be a cuspidal automorphic rep…