11 problems

Let $r_t$ denote the number of zeros of $E_t$ at the central point, and let the one-level density be tested using a function whose Fourier transform is supported in…

Let a family of $L$-functions have symmetry type unitary, symplectic, or even orthogonal. Weight the members of the family by their central $L$-values, and consider the resulting o…

Let $\mathcal{F}(K)=\{L_k(s):1\le k\le K\}$, let $M=\log K$, let $f$ be the test function, and let $W_f$, $S_\zeta$, $S_L$, $S_{A'}$, and $S_\Gamma$ be the integrals defined in the…

Let $\mathcal{F}(K)=\{L_k(s):1\le k\le K\}$, let $M=\log K$, and let $R_K(\alpha,\gamma)$ be the ratios-conjecture prediction built from the Euler product $G(\alpha,\gamma)$. For s…

Let $f$ be an even Schwartz function, let $w$ be the weight function, and define the weighted ratio average … For $\operatorname{Re}(2\nu'+\nu)>-1$ and…

Let $p$) be an odd prime and let $d4a5_p(X)$ be the family of $d4a9/pd4a9$ Galois number fields of discriminant between $X$ and $2X$. Let $d4a5(K,f)$ denote the one-level density a…

One-level density conjecture. The density conjecture states that

The biquadratic-curve one-level density conjecture. As $g\to\infty$,

Let $q$ be fixed, let ${\mathcal{H}_g}$ denote the family of hyperelliptic curves of genus $g$, and let $\Theta_c$ be the associated unitary symplectic matrix. For an even test fun…

Let $\phi(x)$ be a Schwartz-class function on $\mathbb{R}$ whose Fourier transform has compact support. Define … For $f\in H_k$, let $\Lambda(s,f)$ be its completed $L$-function, a…

Let $C$ be the conductor of a natural family of $L$-functions, and consider the one-level density of its zeros at scale $\frac{2\pi}{\log C}$. Iwaniec–Luo–Sarnak's conjecture. In e…