13 problems

Let $F$ be a linear combination of functions in the Selberg class with no Euler-product decomposition in the region of absolute convergence, and suppose that $F$ satisfies a functi…

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

Smoothed weighted one-level density conjecture. As $T\to\infty$,

Let … where the sum is over the non-trivial zeros $\rho=\frac12+i\gamma$ of the Riemann zeta-function, and let $f$ be a real-valued, even test function. For…

Let $L$ be an element of the Selberg class, and write its nontrivial zeros on the critical line as $1/2+it$. Apart from the central point $s=1/2$, these zeros are considered with t…

Let $F\in\mathcal{S}^{\sharp \flat }(\sigma_0,\sigma_1)$, let $\tau$ be the parameter in the generalized Li coefficients $\lambda_F(n,\tau)$, and consider the half-plane…

Bounce-number conjecture. The bounce number satisfies

Let $Q\in\mathbb{F}_q[T]$ have degree $d\geqslant 2$, and let $\mathcal{F}_Q$ be the family of primitive Dirichlet $L$-functions associated with $Q$. For each…

Let a Dirichlet ordinate be a nonnegative real number that is a $\chi$-ordinate for some Dirichlet character $\chi$. Silberman's consequence of linear independence. There does not…

Given a Dirichlet character $\chi$, call a real number $\gamma$ a $\chi$-ordinate if $L(\sigma+i\gamma,\chi)=0$ for some $0<\sigma<1$. Let $a$ and $b$ be positive real numbers. The…

Let $L(s,\chi)$ range over Dirichlet $L$-functions, with zeros counted according to multiplicity. The simplicity conjecture. Every zero of every Dirichlet $L$-function is simple. T…

Let $\zeta(s)$ be the Riemann zeta function, and consider the positive imaginary parts of its nontrivial zeros. The zeta linear-independence conjecture. These imaginary parts are l…

Let $L(s,\chi)$ range over Dirichlet $L$-functions attached to primitive characters, and consider the nonnegative imaginary parts of their nontrivial zeros. The linear-independence…