14 problems

For $x>0$, let … be the prime-counting function. Gauss's conjecture. As $x\to\infty$, … The conjecture anticipated the Prime Number Theorem, proved independently by Hadamard and de…

Let $\chi\pmod{q}$ be a Dirichlet character, and write $s=\sigma+it$. Generalized Lindelöf hypothesis. For every $\varepsilon>0$, if $\sigma\geq 1/2$ and $|s-1|\geq 1/2$, then … Th…

For a positive real number $\alpha$, let $\lambda(\alpha)$ be the smallest non-negative real such that, for every $\varepsilon>0$ and all sufficiently large $n$, the Lebesgue measu…

Chebyshev's prime-counting conjecture. The difference $\pi(x)-\operatorname{li}(x)$ should be negative for every $x$.

For real $x$ and $t>1$, let $S_{\left\lfloor x/n^t\right\rfloor,x}$ denote the set considered in the paper, with the function defining $S_{f,x}$ taken to be non-piecewise. Sparsity…

For $T\ge 3$, let $\mathcal{L}(T)$ and $\beta=\beta(T)$ be non-decreasing continuous functions satisfying … Let $W=W(x)$ be increasing with $W(x)\gg(\log x)^A$ for a sufficiently l…

Let $\pi(x)$ denote the number of primes not exceeding $x$, and let $x$ tend to infinity. Gauss–Legendre conjecture. The prime-counting function satisfies … This is the prime numbe…

Let $(Z,\kappa,R)$ be an ergodic and aperiodic measure-theoretic dynamical system. A strictly ergodic model of this system is a strictly ergodic topological dynamical system…

Modified prime number theorem.

Prime number theorem conjecture. The function $\pi(x)$ is asymptotic to a function of the size of

Let $G(X,c)$ denote the variance of sums of the von Mangoldt function over reduced residue classes modulo $c$. Friedlander–Goldston conjecture. They conjecture that Hooley's asympt…

Let $\pi(x)=\#\{p\leq x:p\text{ is prime}\}$ denote the number of primes at most $x$. The prime number theorem. As $x\to\infty$, … The prime number theorem is a theorem rather than…

Maximal oscillation conjecture. As $x\to\infty$,

Legendre–Gauss prime number theorem conjecture. The ratio