44 problems

Katz's Sato–Tate conjecture. For every $f\in\mathcal{C}([0,\pi])$,

Let $B$ be a semisimple algebra of dimension $n$ over $\q$. For $a\in \q^*$, let $x\in B^*$ be regular and let $r\geq 2$ be an integer. Writing … so that…

Mesnager's conjecture. For every even $m$, the function $f_{a,b}$ is bent if and only if

For integers $m,n$ and positive integers $c$, define the ordinary Kloosterman sum … where $e(x):=\exp(2\pi i x)$ and $\overline{d}$ denotes the inverse of $d$ modulo $c$. Linnik–Se…

For a prime $p$, let ${\mathbb F}_p$ be the finite field with $p$ elements. For $a\in{\mathbb F}_p^*$, define the Kloosterman sum … where ${\mathbf e}_p(z)=\exp(2\pi i z/p)$, and d…

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…

For $m,n\neq0$, define … and let $r(S,S^*)(x)$ be the Pearson correlation of the sequences $S(m,n;c)$ and $S^*(m,n;c)$ over the dyadic window $W_x=\{N\mid c:x<c\leq2x\}$. Uncorrela…

Selberg–Linnik bound. For every $m,n\neq0$ and every $\varepsilon>0$,

Kloosterman $L^1$ anti-concentration conjecture. For every $0<\alpha<\frac12$, there exists a constant $c_\alpha>0$ such that, for every odd prime power $q$, if…

Let $u1< u2$ be positive integers, let $n$ be a positive integer, let $b$ be an integer coprime to $n$, and let $bar a$ denote the multiplicative inverse of $a$ modulo $n$. Define…

Sharpness conjecture. The number of such solutions is at most

Polynomial exponential-sum conjecture. If $g$ and $h$ satisfy

Let $B_{d,n}(r)$ and $\widetilde{B}_{d,n}(r)$ be the functions appearing in the radial Fourier interpolation formulas for dimensions $d=3,4$, with $n$ an appropriate positive integ…

For integers $m,n$ and a real number $X\geqslant 1$, let … where $e(z)=\exp(2\pi i z)$. Modified Selberg–Linnik conjecture. One should have … This conjecture predicts essentially s…

Additively twisted Linnik–Selberg conjecture. One should have

Let $q=p^f$, let $p$ be a sufficiently large prime, and let $x$ be a regular elliptic element of $\operatorname{GL}_n(\mathbb{F})$ with eigenvalues…

Let $k=\frac12$ or $k=-\frac12$, and let $u$ be a weight $k$ multiplier system on $Gamma0(N)$. Let $S(m,n,c,nu)$ be the corresponding half-integral-weight Kloosterman sum, let…

For positive integers $m,n$ and $x$, let … where $\overline{d}d\equiv1\pmod c$. Sarnak–Tsimerman full-cancellation conjecture. For every $\varepsilon>0$, … This conjecture predicts…

Let ${\mathcal K}_m(a)$ and ${\mathcal K}_m^*(a)$ denote the Kloosterman-sum quantities in the paper, let $\mu(m)$ be the Möbius function, and let $\mathsf {H}_{a,\mu}(M)$,…

Let an $SL(3,\mathbb{Z})$ Poincaré series have Fourier coefficients whose Bruhat expansion contains exponential-sum terms associated with the elements of the Weyl group. A properly…

Let $a$ be an $n\times n$ matrix over $\mathbb{F}_q$, with characteristic polynomial $P_a$ irreducible over $\mathbb{F}_q$, and let $\alpha\in\mathbb{F}_{q^n}$ be an eigenvalue of…

Nonexistence conjecture. The function $F$ does not permute $\mathbb F_{2^n}$.

For positive integers $C$, $D$, and $n$, let $e(D/c)$ denote the additive twist and let $S(n,n,c)$ be the Kloosterman sum. Iwaniec's additively twisted Selberg–Linnik conjecture. F…

In the spectral Kuznetsov formula for $GL(3)$, let the $w_5$ cell and the long-element cell denote the corresponding Weyl-element contributions, and let the integral transforms be…