7 problems

Let $f$ be a half-integral-weight Hecke eigenform, and let $b(n)$ denote its normalized Fourier coefficients. Bruinier–Kohnen conjecture. The signs of the coefficients of $f$ are e…

For each odd square-free integer $t$, let $\Delta(t)=r_{Q_1}(t)-r_{Q_2}(t)$, and define … … and $N_{\ne 0}(X)=N_+(X)+N_-(X)$. Sign equidistribution conjecture. As $X\to\infty$, … E…

Jutila-type gap conjecture. For every $\epsilon>0$, there exists $T_0(\epsilon)$ such that, for all $T\geq T_0(\epsilon)$, $L(s,f)$ has a zero of the form

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…

Let $f$ be as above, let $I\subseteq\mathbb R_{>0}$ be any interval, and let $b(n)$ be the normalized coefficients of $f$. Independence of sign and absolute value conjecture. The f…

Let $g\in S_{k+\frac12}^+$ be a general cusp form with Fourier coefficients $c(n)$, and let $D$ be a fundamental discriminant. For every $\varepsilon>0$, the conjecture concerns th…

Let $k$ be a positive integer, $N$ a positive integer, and let $\chi$ be a Dirichlet character modulo $4N$. Suppose that $f\in M_{k+1/2}(4N,\chi)$ is a Hecke eigenform, so that Shi…