|
VIDEO LIBRARY |
Conference to the Memory of Anatoly Alekseevitch Karatsuba on Number theory and Applications
|
|||
|
On a discrete spectrum of Laplace operator on the fundamental domain of modular group and Chebyshev's function D. A. Popov Lomonosov Moscow State University, Belozersky Research Institute of Physico-Chemical Biology |
|||
Abstract: Laplace operator $\Delta = u^{2}\bigl(\partial_{x}^{2}\,+\,\partial_{u}^{2}\bigr)$ has infinite discrete spectrum $$ \Delta\varphi_{n}\,+\,\lambda_{n}\varphi_{n}\,=\,0,\quad \varphi_{n}\in L^{2}(\mathcal{F},d\mu),\quad d\mu\,=\,\frac{dx\,du}{u^{2\mathstrut}},\quad\lambda_{n}\ge 0, $$ in fundamental domain $$ \mathcal{F}\,=\,\bigl\{z\,=\,x+iy\,|\,y>0, |z|>1, |x|<\tfrac{1}{2}\bigr\} $$ of modular group In his “Shur Lectures” (Tel -Aviv, 1992), P. Sarnack suggested that this discrete spectrum In the talk, we will discuss a proof of the following new theorem: Theorem. Let $$ 0<t\le x^{-4}(\ln{x^{p}})^{-2},\quad p\ge 20. $$ 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 Therefore, the fact that the discrete spectrum Language: Russian and English |