9 problems

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 $m_1,m_2$ be integers of the same parity. Vatwani's Gaussian prime pair conjecture. There are infinitely many pairs of rational primes $(p_1,p_2)$ of the form … The conjecture…

Gaussian Primes Conjecture. One has

Gaussian-prime zero-statistics form-factor conjecture. As $K\to\infty$,

Smooth Gaussian-prime variance conjecture. The variance satisfies

Gaussian-prime angle variance conjecture. For $1\ll K\ll N^{1-o(1)}$,

Let $a+bi$ be a Gaussian prime and let $\frac{q}{p}$ be the Legendre symbol appearing in the comparison of the two Gaussian-prime ratios $\frac{a+bi}{\beta+\beta i}$ and…

Number-of-moduli conjecture. There exists a constant $C>0$, depending only on $k$ and $\ell$, such that