# 塞尔伯格迹公式

## 定义

${\displaystyle X}$ 为紧致、负常曲率曲面，这类曲面可以表为上半平面 ${\displaystyle \mathbb {H} }$${\displaystyle \mathrm {PSL} (2,\mathbb {R} )}$ 的某离散子群 ${\displaystyle \Gamma }$ 的商。

${\displaystyle X\ =\ \Gamma \backslash \mathbb {H} }$

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

${\displaystyle -\ \Delta \ u_{n}(x,y)\ =\ \lambda _{n}\ u_{n}(x,y)}$

${\displaystyle 0=\lambda _{0}<\lambda _{1}\leq \lambda _{2}\leq \cdots }$

## 迹公式

${\displaystyle \sum _{n=0}^{\infty }h(r_{n})={\frac {\mu (F)}{4\pi }}\int _{-\infty }^{\infty }r\,h(r)\,\tanh(\pi r)\,dr\ +\ \sum _{\{T\}}{\frac {\log N(T_{0})}{N(T)^{1/2}-N(T)^{-1/2}}}\ g(\log N(T))}$

• 在带状区域 ${\displaystyle \vert \Im \mathrm {m} (r)\vert \leq 1/2+\delta }$ 上为解析函数，在此 ${\displaystyle \delta >0}$ 为某常数。
• 偶性：${\displaystyle h(-r)=h(r)}$
• 满足估计：${\displaystyle \vert h(r)\vert \leq M\ \left(1+\vert \Re \mathrm {e} (r)\vert ^{-2-\delta }\ \right)}$，在此 ${\displaystyle M>0}$ 为某常数。

${\displaystyle h(r)=\int _{-\infty }^{\infty }g(u)\ e^{iru}\ du}$

## 文献

• 叶扬波《模形式与迹公式》，北京大学出版社，2001年。 ISBN 7-301-04586-7
• A. Selberg, Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces With Applications to Dirichlet Series, Journal of the Indian Mathematical Society 20 (1956) 47-87.
• H.P. McKean, Selberg's Trace Formula as Applied to a Compact Riemannian Surface, Communications in Pure and Applied Mathematics 25 (1972) 225-246. 勘误见 : Communications in Pure and Applied Mathematics 27 (1974) p.134
• D. Hejhal, The Selberg Trace Formula and the Riemann Zeta Function, Duke Mathematics Journal 43 (1976) 441-482
• D. Hejhal, The Selberg Trace Formula For PSL(2,R), Volume I, Springer Lecture Notes 548 (1976), ISBN .
• A.B. Venkov, Spectral Theory of Automorphic Functions, the Selberg Zeta Function, and Some Problems of Analytic Number Theory and Mathematical Physics, Russian Mathematical Surveys 34 (1979) 79-153.
• P. Cartier and A. Voros, Une Nouvelle Interprétation de la formule des traces de Selberg, dans The Grothendieck Festschrift, volume 87 of Progress in Mathematics, Birkhäuser (1990) 1-67.
• （英文） Matthew R. Watkins, Selberg trace formula and zeta functions