On a discrete spectrum of Laplace operator on the fundamental domain of modular group and Chebyshev's function

Laplace operator $\Delta = u^{2}\bigl(\partial_{x}^{2}\,+\,\partial_{u}^{2}\bigr)$ has infinite discrete spectrum $\{\lambda_{n}\}$,
$$\Delta\varphi_{n}\,+\,\lambda_{n}\varphi_{n}\,=\,0,\quad \varphi_{n}\in L^{2}(\mathcal{F},d\mu),\quad d\mu\,=\,\frac{dx\,du}{u^{2\mathstrut}},\quad\lambda_{n}\ge 0,$$
in fundamental domain
$$\mathcal{F}\,=\,\bigl\{z\,=\,x+iy\,|\,y>0, |z|>1, |x|<\tfrac{1}{2}\bigr\}$$
of modular group $PSL(2,\mathbb{Z})$. Also, it has a continuous spectrum mapping the interval$\bigl[\tfrac{1}{4},+\infty\bigr)$.
In his “Shur Lectures” (Tel -Aviv, 1992), P. Sarnack suggested that this discrete spectrum $\{\lambda_{n}\}$ should play a key role in number theory.
In the talk, we will discuss a proof of the following new theorem:

Theorem. Let $x\ge 3$ and suppose that
$$0<t\le x^{-4}(\ln{x^{p}})^{-2},\quad p\ge 20.$$
Then the following formula holds:
$$\psi(x)\,=\,2\sqrt{\pi}t\sum\limits_{n\ge 0}e^{-tr_{n}^{2}}\sum\limits_{2\le k\le x}k\cos{(2r_{n}\ln{k})}\,+\,R(x),\quad |R(x)|\,\le\,\frac{cx^{2}\sqrt{t}}{(\ln{x})^{2\mathstrut}}\,\le\,\frac{c}{(\ln{x})^{3\mathstrut}}.$$
\emph{Here $\psi(x)$ denotes Chebyshev's function, the sequence $r_{n}$ is defined by the relations $\lambda_{n}=r_{n}^{2}+\tfrac{1}{4}$, and $c$ is some computable constant.}

Therefore, the fact that the discrete spectrum $\{\lambda_{n}\}$ determine prime number distribution, is established.