29 problems

Let $b(n)$ be the indicator of the positive integers representable as sums of two squares, and let $q$ be a fixed positive integer. Connors–Keating conjecture. There is a precise p…

Infinitude conjecture. $E^*_{Y^2}$ contains infinitely many numbers of the form $\pi_1\pi_2$ for such primes $\pi_1$ and $\pi_2$.

Quadratic-scaling pair correlation conjecture. A nonzero asymptotic pair correlation density exists for the quadratic scaling. Moreover, it exhibits strong level repulsion and vani…

Let $\mathbf{p}(x)\in\mathbb{Z}[x]$ be an irreducible monic polynomial of degree $3$. An integer-valued quadratic polynomial $q_{0}(x)$ is a quadratic polynomial taking integer val…

Let $D=(0,\pi)\times(0,\pi)$, and consider the Dirichlet Laplacian eigenvalue problem … Its eigenvalues are $\lambda_{nm}=n^2+m^2$ for $n,m\in\mathbb{N}$, with multiplicity determi…

Let $r(n)$ denote the number of ways to write $n$ as a sum of two squares of integers, and let … It is known that $R(x)=\pi x+O(x^\theta)$ for some $\theta\in(\frac14,\frac12)$. Ci…

Schmutz Schaller's conjecture. For every real number $x\ge 1$,

Let $k\in\mathbb{Z}_{\geqslant 2}$ and $x\to\infty$. Let $(m_i,n_i)\in\mathbb{Z}^2$ for $1\leqslant i\leqslant k$, with $(m_i,n_i)\ne(m_j,n_j)$ for $i\ne j$. An admissible represen…

Hooley's short-interval conjecture. The asymptotic

Fix a prime $q\equiv1\,(\mathrm{mod}\,4)$, an integer $r\geq2$, and $\mathbf{a}=(a_1,\dots,a_r)\in\mathbb{N}^r$. Let $E_n$ be the increasing sequence of sums of two squares, and de…

For $k\geq1$, let $\mathcal{H}=\{h_1,\dots,h_k\}\subseteq\mathbb{Z}$, let $q\equiv1\,(\mathrm{mod}\,4)$ be prime, let $a\in\mathbb{Z}$, and let $R_k(\mathcal{H};x,q,a)$ and…

For $k\geq 1$, let $\mathcal{H}=\{h_1,\dots,h_k\}\subseteq\mathbb{Z}$, let $q\equiv1\,(\mathrm{mod}\,4)$ be prime, let $a\in\mathbb{Z}$, and let $R_k(\mathcal{H};x,q,a)$ be the nor…

Let $\mathcal{H}^*=\{h_1,h_2,\ldots\}$ be an ordered set of integers. It is admissible if every finite truncation $\{h_1,\ldots,h_k\}$ is admissible, meaning that its reduction mod…

Let $b(n)$ be the indicator of sums of two squares. For an odd prime $p$, the Connors–Keating correlation conjecture specializes to the following explicit local factors. Connors–Ke…

Let $V_b(X;H)$ be the variance defined from the error in the smooth approximation to counts of sums of two squares in intervals of length $H$. The order-of-magnitude variance conje…

Let $b(n)$ be the indicator of integers representable as sums of two squares. Let $B(x)=\sum_{n\leq x}b(n)$, let $I(x;H)$ be the paper's smooth approximation to $B(x+H)-B(x)$, and…

Gap conjecture for sums of two squares. For every $\varepsilon>0$,

Let $Q(x,y):=Ax^2+2Bxy+Cy^2+Dx+Ey+F$ be a binary quadratic form with integral coefficients satisfying … and with discriminant … for a prime $q=4k+1$. Suppose that $Q(x,y)$ is posit…

Let $M_{n,q}$ be the monic degree-$n$ polynomials in $\mathbb{F}_q[T]$, let $1_{\mathbb{P}}(f)$ indicate that $f$ is prime, and let $b_q(f)$ indicate that $f=A^2+TB^2$ for some…

Let $M_{n,q}$ be the monic degree-$n$ polynomials in $\mathbb{F}_q[T]$, let $b_q(f)$ indicate that $f=A^2+TB^2$ for some $A,B\in\mathbb{F}_q[T]$, and let…

Function-field autocorrelation conjecture. Uniformly as $q^n\to\infty$,

Freiberg–Kurlberg–Rosenzweig's conjecture. The higher autocorrelation satisfies

Let $\mathbb{S}$ be the set of integers expressible as a sum of two squares. For $k\geq1$ and a set $\boldsymbol{h}=\{h_1,\ldots,h_k\}\subseteq\mathbb{Z}$ with…

Let $\mathbb{S}$ be the set of integers expressible as a sum of two squares. For $k\geq 1$ and a set $\boldsymbol{h}=\{h_1,\ldots,h_k\}\subseteq\mathbb{Z}$ with…

Let $\mathbb{S}$ denote the set of integers expressible as a sum of two squares. For $k\geq 1$ and a set $\boldsymbol{h}=\{h_1,\ldots,h_k\}\subseteq\mathbb{Z}$ with…