12 problems

Heath-Brown's conjecture. One has

Let $\psi(X;Q,A)=\sum_{\substack{n\leq X\\ n=A\bmod Q}}\Lambda(n)$, where $\Lambda$ is the von Mangoldt function, and define … In the trivial regime $X<Q$, Hooley's conjecture. Und…

Let $\psi(x;k,a)$ denote the Chebyshev function for primes in the arithmetic progression $a\pmod{k}$, and let $\varphi$ denote Euler's totient function. Friedlander–Granville's con…

Let $q$ and $\ell$ be integers such that $1\leq\ell\leq q$ and $(\ell,q)=1$. Define $M(q,\ell)$ by … Let $\delta_{\ell}=1$ if $\ell$ is prime and $\delta_{\ell}=0$ otherwise. Langu…

Xylouris's least-prime bound. For any relatively prime $c$ and $d$, there exists a prime $p$ congruent to $c$ modulo $d$ such that

For $0<\theta\leqslant1$, let $\psi(x;q,a)$ denote the weighted prime-counting function in the arithmetic progression $a\pmod q$, and let $\varphi(q)$ be Euler's totient function.…

Let $E$ be an elliptic curve over ${\mathbb Q}$, let $f$ and $a$ be relatively prime positive integers, and let $K_d={\mathbb Q}(E[d])$ be the $d$-division field. For a primitive…

For real $x$, a positive integer modulus $q$, and an integer residue $r$, let $\pi(x;q,r)$ denote the number of primes less than $x$ that are congruent to $r$ modulo $q$. Prime pro…

Let $a$ and $d$ be coprime integers with $1\leq a<d$, and let the first prime in the arithmetic progression $a,a+d,a+2d,\ldots$ be bounded above by $cd^L$, for positive constants…

Let $\mathbb{F}_q[T]$ be the polynomial ring over the finite field $\mathbb{F}_q$, let $\Lambda_q$ be its von Mangoldt function, and define … For nonconstant…

Let $q$ and $d$ be relatively prime positive integers, and consider the cyclotomic extension $L=\mathbb{Q}(\mu_q)$ with Galois group $G=(\mathbb{Z}/q\mathbb{Z})^\ast$. For…

Let $p(r,a)$ denote the least prime in the arithmetic progression $a$ modulo $r$. Here $r$ is a power of $2$ and $a=1$. Prime progression conjecture. There exists a constant $c>0$…