3 problems

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 $\Gamma_0(N)$. Let $S(m,n,c,\nu)$ be the corresponding half-integral-weight Kloosterman sum, let…