23 problems

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 $B$ 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…

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…

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

Additively twisted Linnik–Selberg conjecture. One should have

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…

Fix an integer $a\ne0$, positive integers $\nu_1,\ldots,\nu_s$, and integers $h_s>\cdots>h_1\geqslant0$. Let ${\mathcal K}_m^*(a)$ be the normalized Kloosterman-sum quantity and le…

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…

Let $K=\mathbb{F}_q(t)$, $K_{\infty}=\mathbb{F}_q(!(1/t)!)$, and let $\psi$ be the nontrivial additive character on $K_{\infty}$ that is trivial on $\mathbb{F}_q[t]$. For nonzero…

Let $p$ be prime, let $a\in\mathbb{F}_p^\times$, let $N$ and $H$ be integers, and write $\overline{x}$ for the multiplicative inverse of $x$ modulo $p$. Short sums conjecture. Ther…

Let $f$ be a fixed primitive cusp form, holomorphic or Maass, and let $\lambda_f(n)$ denote its Hecke eigenvalue at $n$. Write $\mathcal{L}=\log X$. Integer correlation conjecture.…

Let $f$ be a fixed primitive cusp form, holomorphic or Maass, and let $\lambda_f(p)$ denote its Hecke eigenvalue at a prime $p$. Write $\mathcal{L}=\log X$. Prime correlation conje…

Bentness conjecture. The function $f_{a,b}$ is bent if and only if $K_{n/2}(a)=4$.

Let … where $M\leq m<2M$, $N\leq n<2N$, $A\leq a<2A$, and $\|\cdot\|$ and $\|\cdot\|_\infty$ denote the $L_2$ and $L_\infty$ norms. The conjecture concerns the range…

Let $q$ be prime, let $a$ be an integer coprime with $q$, let $\boldsymbol{b}$ lie in the generic set … and let $[?]$ be elements of $[?]$. For the correlation sum $[?]$ formed usi…

Kloosterman-sum identity conjecture.