7 problems

Goldston–Montgomery variance conjecture. For every fixed $0<\delta<1$, this variance satisfies

Let $N$ tend to infinity, and let $H$ be the length of the interval in the prime-counting function. The distribution of the normalized values of $\psi(n+H)-\psi(n)$ for $n\in[1,N]$…

Let $\Lambda(n)$ be the von Mangoldt function, and for distinct integers $d_1,\ldots,d_k$ with $1\leq d_i\leq H$, let $\mathcal{D}=\{d_1,\ldots,d_k\}$. Define $E_k(x;\mathcal{D})$…

Let $N$ be large and let $\epsilon>0$ be fixed. Suppose that … Define the von Mangoldt function by … and set … Let $\mu_k$ denote the $k$th moment of the standard normal distributi…

Let $\psi(x):=\sum_{n\leq x}\Lambda(n)$ be the Chebyshev function. For fixed $\varepsilon>0$ and $n\in\mathbb N$, consider $\delta$ uniformly in the range … Let $\mu_n$ denote the…

Let $\psi(x)=\sum_{n\leqslant x}\Lambda(n)$ be the Chebyshev function. Short-interval variance conjecture. For all $\varepsilon>0$, if $h\leqslant X^{1-\varepsilon}$, then … as…

Conjectured unification. The estimate