24 problems

Assume the Riemann Hypothesis and let $\gamma,\gamma'$ range over imaginary parts of non-trivial zeros of $\zeta(s)$. For $0<\alpha<\beta\ll\log T$, the short-interval pair-correla…

Assume the Riemann Hypothesis, let $\gamma$ and $\gamma'$ range over imaginary parts of non-trivial zeros of $\zeta(s)$, and let $h$ be the shift parameter. For fixed $\alpha>0$, t…

Let $\rho=\beta+i\gamma$ and $\rho'=\beta'+i\gamma'$ range over the nontrivial zeros of the Riemann zeta function. For fixed $\lambda>0$ and $T\to\infty$, consider pairs with posit…

Fix $\theta\in[0,2\pi)$, let $a=1/2-c/\log T$ with $0<c=o(\log T)$, and let $A>0$ be arbitrarily large but fixed. Let $f_a(w,\theta)$ be the function in the source, and let…

Twisted pair correlation conjecture. Fix $\varepsilon>0$ and let $r\in\mathfrak C(\mathbb R)$. Uniformly for all prime powers $n=q^a\leq T^{1-\varepsilon}$,

Strong twisted pair correlation conjecture. Fix $\varepsilon>0$. Uniformly for all prime powers $n\leq T^{1-\varepsilon}$,

Pair correlation for Dirichlet $L$-functions. Under GRH, as $x\to\infty$,

Let $\alpha\neq 0$ and let $\theta>0$ satisfy $\theta\neq\tfrac12$ and $\theta\notin\mathbb Z$. For each $N$, let $y_1^{(N)}<\cdots<y_N^{(N)}$ be an ordering of the points…

Let $F_\Delta(\alpha)$ denote the shifted pair-correlation function and let $w(\Delta)=4/(4+\Delta^2)$. Chan's extended strong pair correlation conjecture. For $|\alpha|\geq1$ and…

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…

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 $D<0$ be a fundamental discriminant, let $H(D)$ be the ideal class group of $\mathbb{Q}(\sqrt{D})$, and let $\hat{H}(D)$ be the family of unramified Hecke characters associated…

Let $\left(a_n\right)_{n\geq 1}$ be a strictly increasing sequence of positive integers. It is quasi-arithmetic of degree $d$ if there exist constants $C,K>0$ and a strictly increa…

Let $\left(a_n\right)_{n\geq 1}$ be a sequence of distinct integers, and let $A_N$ denote its first $N$ elements. The additive energy of a finite set $A$ of reals is … where the su…

Let $\mathcal{P}$ be an Apollonian gasket and let $\mathcal R\subset\mathbb C$ satisfy $\mu(\mathcal R)>0$. For $T>0$, let $F_{T,\mathcal R}(s)$ denote the pair correlation functio…

Let $\mathbb{H}(\mathbb{R})$ be the Heisenberg group, let $\mathbb{H}(\mathbb{Z})$ be its integer subgroup, and let $\mu_H$ be the invariant probability measure on…

Let $\varphi(z)=\alpha_q z^q+\alpha_{q-1}z^{q-1}+\cdots+\alpha_0$ be a polynomial of degree $q$, and let $P_N(\,\cdot\,;\varphi)$ denote the gap distribution of…

Pair-correlation density conjecture. The pair-correlation measure $R_2(\xi)$ exists on $[0,\infty)$ and is given by a $C^1$ function expressed as a series of three-dimensional volu…

For each Dirichlet character $\chi$ modulo $q$, let $\gamma_\chi$ and $\gamma'_\chi$ denote ordinates of nontrivial zeros of $L(s,\chi)$, counted with multiplicity. Fix $C>0$. Pair…

Technical divisor-sum conjecture. Then

Rudnick–Sarnak–Heath-Brown conjecture. Assume $\beta$ is Diophantine. Then the pair correlation for the fractional parts of $n^2\beta$ is Poissonian.

Let $\delta\in(0,1)$ be fixed, let $q$ and $c$ be integers with $(c,q)=1$, and let $N$ be positive. Uniform multiplication-table congruence conjecture. One has … uniformly for…

Let $F\in\mathcal S$ and let $\mathcal D_F(x,T)$ be the normalized pair-correlation function defined above. Smooth-variation conjecture. For every $A>0$, there exists $c>0$ such th…

Murty–Perelli pair-correlation conjecture.