Selberg's 1/4 conjecture

Let $\mathbb{H}=\{z=x+iy\in\mathbb{C}: y>0\}$ be the upper half-plane, equipped with the hyperbolic Laplace–Beltrami operator

$$\Delta \;=\; -y^{2}\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right),$$

and let $\operatorname{SL}_2(\mathbb{Z})$ act on $\mathbb{H}$ by fractional linear transformations $z\mapsto \frac{az+b}{cz+d}$. For an integer $N\ge 1$ let

$$\Gamma(N)=\left\{\begin{pmatrix} a & b\\ c& d\end{pmatrix}\in \operatorname{SL}_2(\mathbb{Z}) \;:\; \begin{pmatrix} a & b\\ c& d\end{pmatrix}\equiv \begin{pmatrix} 1 & 0\\ 0& 1\end{pmatrix} \ (\mathrm{mod}\ N)\right\}$$

be the principal congruence subgroup of level $N$, and more generally call a subgroup $\Gamma\subseteq \operatorname{SL}_2(\mathbb{Z})$ a congruence subgroup if $\Gamma\supseteq\Gamma(N)$ for some $N\ge 1$.

For a congruence subgroup $\Gamma$, a Maass cusp form of eigenvalue $\lambda\in\mathbb{C}$ for $\Gamma$ is a smooth function $f:\mathbb{H}\to\mathbb{C}$, not identically zero, such that

$$f(\gamma z)=f(z)\quad \text{for all } \gamma\in\Gamma,\ z\in\mathbb{H},\qquad \Delta f=\lambda f,$$

and such that $f$ is square-integrable on $\Gamma\backslash\mathbb{H}$ with respect to the hyperbolic measure $y^{-2}\,dx\,dy$ and tends to $0$ in each cusp of $\Gamma$ (equivalently, $f$ has vanishing constant term in its Fourier expansion at every cusp).

Then for every congruence subgroup $\Gamma\subseteq\operatorname{SL}_2(\mathbb{Z})$, every eigenvalue $\lambda$ of a Maass cusp form for $\Gamma$ satisfies

$$\lambda \;\ge\; \tfrac14 .$$

Equivalently, writing $\lambda_1(\Gamma)$ for the smallest positive eigenvalue of $\Delta$ in the discrete spectrum of $L^{2}(\Gamma\backslash\mathbb{H})$, one has $\lambda_1(\Gamma)\ge \tfrac14$; in the parametrization $\lambda=s(1-s)$ with $s=\tfrac12+it$, every such $\lambda$ has $t\in\mathbb{R}$.