10 problems

Jutila's conjecture. For all $k=1,2,\dots$ and $X\geq 3$, $Y\geq 1$, there are coefficients $c_1(k)$ and $c_2(k)$ depending only on $k$ such that

Let $k$ be a positive integer, let $r$ be a positive fundamental discriminant, and let $\chi_r$ be the primitive quadratic character modulo $r$. Define … For $y>0$, define the vari…

Let $\mathcal{D}(X)$ be the set of fundamental discriminants $d$ with $|d|<X$, and let $L(s,\chi_d)$ be the Dirichlet $L$-function attached to the quadratic character $\chi_d$. For…

Let $\EuScript F_{\EuScript D}$ be the family of real primitive quadratic Dirichlet characters $\chi_d$, with $d$ nonzero and squarefree, and let $r\geq4$. For integers $N\geq1$ an…

The logarithmic-derivative conjecture. Then

Let $\varphi(x)$ be a Schwartz class function on $\mathbb{R}$ with Fourier transform having compact support. Define the density … and let $\rho=\frac12+i\gamma$, with…

Diaconu–Goldfeld–Hoffstein conjecture. As $D\to\infty$,

Average first negative value conjecture. As $x\to\infty$, one expects

CFKRS–DGH twisted moment conjecture. Uniformly for $|\operatorname{Re}(\alpha)|\ll(\log X)^{-1}$ and $|\operatorname{Im}(\alpha)|\ll(\log X)^2$, one has

CFKRS–DGH conjecture. The first moment satisfies