108 problems

Generalized Xi identity conjecture. For all $s_0\in\mathbb C$ and all $a\in\mathbb C$,

General constraint conjecture.

Let $n$, $a$, and $d$ be positive integers, and set … Let $\varphi(n)$ denote Euler's totient function and let $\zeta_k$ be a primitive $k$-th root of unity. Arithmetic conjecture.…

For integers $r\ge 1$ and real numbers $n\ge 1$, $0<\eta\le 1/2$, define … Here $a_i,b_j\sim n$ denotes the same dyadic range as in the source, and $\lVert\cdot\rVert$ is distance…

Almost-everywhere Weyl-sum exponent conjecture. For $\alpha\in[0,1]$,

Let $\Delta$ be a triangle with vertices at $(0,0)$, $\mathbf P_1=(a_1,b_1)$, and $\mathbf P_2=(a_2,b_2)$. Assume the hypotheses on the prime $p$ from the main theorem. Let…

For $\alpha>0$, let $A(\alpha)$ be the asymptotic constant conjectured for … and let $B(\alpha)$ be the corresponding constant when $|z_k|=1$. Equality of asymptotic constants conj…

Exact unit-circle asymptotic conjecture. One can choose $B(\alpha)=\underline B(\alpha)=\overline B(\alpha)$ in Theorem $avs$. This conjecture would close the gap between the known…

Exact asymptotic formula conjecture. One can choose $A(\alpha)=\underline A(\alpha)=\overline A(\alpha)$ in Theorem $u77$. This would determine the true asymptotic constant for eve…

Vanishing conjecture. $H_k^0 = 0$ for all positive integers $k$. Consequently, $M_k(T)$ is a polynomial for all positive integers $k$.

Igusa's conjecture. There exists a constant $c$ such that for every prime $p$ and every positive integer $m$,

Let $a,q\geq2$ be integers with $(a,q)=1$ and $q$ not a perfect square. Let $H,K\geq1$ and $\lambda,\mu$ be reals. Define $e(u):=e^{2\pi i u}$, let…

Let $a,q\geq2$ be integers with $(a,q)=1$ and $q$ not a perfect square, and let $H,K\geq1$ be reals. Define $e(u):=e^{2\pi i u}$, let $\left(\frac{\cdot}{\cdot}\right)$ denote the…

Let $m\geq 3$ be odd, let $n$ be a non-negative integer, and let $f$ be a quadratic polynomial in $n$ variables modulo $m$. Write $q=2\cos\left(\frac{\pi}{2m}\right)$ and let…

Let $m\geq 3$ be odd, let $n$ be a non-negative integer, and let $f$ be a quadratic polynomial in $n$ variables modulo $m$. Write $q=2\cos\left(\frac{\pi}{2m}\right)$ and let…

Mean-value formula conjecture. The mean value is equal to

Let $p$ be prime and define … Here $a$ is understood modulo $p$, $k\in\mathbb{Z}_p^\times$, and $M$ is in the range used for incomplete sums in the paper. Uniform power-sum conject…

Wan's conjecture. There is a Zariski dense subset ${\mathcal{U}}$ in $\mathbb{A}^d$ such that, for every $f(x)\in {\mathcal{U}}$, the limit exists and

For a finite field $\mathbb{F}_{q}$, let $\mathcal{C}$ be the multiplicative group $\mathbb{F}_{q}^{*}$ or the group of norm-one elements in a quadratic extension, and let…

Wan's conjecture. For $a\in\mathbb F_q^{\times}$, regular $x\in B^{\times}$, and integer $r\ge2$, one has

Let $r\geq 1$ be fixed. For $h\sim H$, define the moment $M_{2r}(H,n)$ as in the source. Ideal moment conjecture. For every fixed $\delta>0$ and every $\varepsilon>0$, one expects…

Weighted dual quadratic large sieve conjecture. For every $\varepsilon>0$,

Let $A_N(\gamma)$ be the oscillatory phase average associated with the centred layer moment, let $X$ be the relevant scale, and let $R$ be the corresponding correlation quantity. S…

Let $11\le p\le 12$. Assume $a\ge b\ge 0$ and $a+b=\frac{p}{2}-3$. Let … and let $\Phi(t)=(t,t^2,t^3,t^4)$ be the moment curve in $\mathbb{R}^4$. Here $n\sim N$ denotes summation o…

Let $T(m,n;c)$ denote the Salié sum for odd positive $c$, and let $\theta$ be the inverse of $L$ modulo $c$ when it exists. For every $\epsilon>0$ there is a $\delta>0$ such that, wh…