45 problems

Let $\alpha\neq 0$ and let $\theta>0$ satisfy $\theta\neq\tfrac12$ and $\theta\notin\mathbb Z$. For $s>0$, define the pair-correlation statistic by … where $\|x\|$ denotes the dist…

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…

Khintchine-type threshold conjecture. For almost all $\alpha$, the sequence $(\{a_n\alpha\})_{n\geq 1}$ has PPC if and only if

Let $F_h(\alpha)$ and $F(\alpha)$ denote the functions associated with the pair correlation of zeros used in the paper, let $h$ be the interval parameter, let $L=\log T$, and let…

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…

Assume the Riemann Hypothesis. Let $F_h(x,T)$ be the shifted pair-correlation function studied in the paper, let $F_h(\alpha):=(\frac{T}{2\pi}\log T)^{-1}F_h(T^\alpha,T)$ for real…

Let $F(x,T)$ be the weighted pair-correlation sum over imaginary parts $\gamma,\gamma'$ of non-trivial zeros of the Riemann zeta function, with … For any small $\epsilon>0$ and any…

Let $\rho_{\beta}^{(2)}(0,\lambda)$ denote the pair correlation function of the Sine-$\beta$ process, with $\beta>0$ and $\lambda$ the separation parameter. For even integer…

For a positive integer $\mu$, let ${\mathfrak Z}_\mu(T)$ denote the set of $\mu$-tuples of zero ordinates under consideration, let…

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 $n\leq T^{1-\varepsilon}$,

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

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

Let $\alpha$ be a real number, and consider the sequence $(\{\alpha n^2\})_{n\ge 1}$ of fractional parts. A real number is Diophantine if it satisfies a Diophantine approximation c…

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 $\mathcal F_{N,k}$ be the family of forms under consideration, let $R_2(g,\rho)(f)$ denote its pair correlation function, and let $x$ tend to infinity. Write $\mathbb E$ for av…

Let $F(x,T)$ be Montgomery's pair correlation function for zeros of the Riemann zeta-function. Feng–Soundararajan–Zhang's conjecture. One expects an asymptotic formula for $F(x,T)$…

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 $F(x,T)$ be Montgomery's pair correlation function for the zeros of the Riemann zeta-function, and let $M$ be any fixed positive number. Heath-Brown's conjecture. Assuming the…

Let the pair correlation density function refer to the density governing the pair correlations of the imaginary parts of zeros of the Riemann zeta-function. Gallagher–Mueller conje…

Aistleitner–El-Baz–Munsch conjecture. If an increasing sequence of reals satisfies

Assume the Generalized Riemann Hypothesis for Dirichlet $L$-functions. Let $\Phi$ be a real-valued compactly supported function satisfying $\Phi(x^{-1})=\Phi(x)$,…

Let $(a_n)_{n\geq 1}$ be a sequence of positive integers, and let $(\{a_n\alpha\})_{n\geq 1}$ denote the associated sequence modulo one. The source asks whether there can be a sequ…