8 problems

Knapowski–Turán conjecture. The proportion of positive real numbers lying in $S$ tends to $1$ as $x\to\infty$. This is an early formulation of Chebyshev's bias, asserting that prim…

Let $q$ be a positive integer, let $\chi$ range over the non-principal Dirichlet characters modulo $q$, and let $L(s,\chi)$ be the associated Dirichlet $L$-function. Define … the o…

Chebyshev-like bias conjecture. (Weak) If $U$ is a regular Lucas sequence with $D>0$, then

Shanks' higher-order bias conjecture. The average of $\tau^{(a)}(n)$ is $(-1)^a$. This extends Chebyshev's bias to integers with a prescribed number of prime factors; the supplied…

Shanks' normalized bias conjecture.

Let $\lambda$ be the Liouville function on the positive integers, and let $\chi_{-4}$ be the nontrivial quadratic character modulo $4$. Shanks' bias conjecture. There are more posi…

Deep Riemann hypothesis. The limit

Let $Q=\{q:q=m^2+1\text{ is prime}\}$, and define the logarithmic densities … … where $\pi_q(n;3,a)$ counts elements of $Q$ up to $n$ that are congruent to $a$ modulo $3$. No-bias…