108 problems

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.…

Let $f\in\mathbb{Z}[x,y]$, let $C_p$ be the curve modulo $p$ defined by $f(x,y)\equiv0\pmod p$, and write $e_{p^2}(u)=\exp(2\pi i u/p^2)$. Exponential-sum conjecture. There exists…

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

General constraint conjecture.

Let $\phi_3,\phi_4$ be real analytic functions on $(0,3)$ satisfying the derivative bounds and nondegeneracy conditions … … … For a finite interval $I\subset {\mathbb Z}$, define ……

Let $p$ run through the primes with $p\equiv 1\pmod 3$, and define … where $e(y)=e^{2\pi i y}$. Let … where $\Gamma$ is the classical gamma function. Patterson's conjecture. As…

Let $\Delta$ be a lattice polytope of degree denominator $D$, and let ${\rm GNP}(\Delta,p)$ and ${\rm HP}(\Delta)$ denote its generic Newton polygon and Hodge polygon, respectively…

Let $d\geq 3$ be an integer, let $N\in\mathbb N$, let $a_1,\ldots,a_N\in\mathbb C$, and write $\mathbb T^2$ for the two-dimensional torus. For $p\geq 2(d+1)$ and every…

Let $f$ be a non-constant monic polynomial in $Q[x]$. A global permutation polynomial (GPP) over $Q$ is a polynomial $P(x)\inQ[x]$ whose reduction modulo $p$ is a permutation of…

Kloosterman-sum identity conjecture.

Let $C_f(s,T)$ be the $C$-function of the $T$-adic exponential sums associated with a polynomial $f$, and let its absolute $T$-adic Newton polygon mean the $T^a$-adic Newton polygo…

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…