Statistical Properties of Eigenvalues of the Hecke Operators
Statistical Properties of Eigenvalues of the Hecke Operators
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DOI：
10.1007/978-1-4612-4816-3_19
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发表时间：
1987
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影响因子：
--
通讯作者：
P. Sarnak
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文献类型：
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作者：
P. Sarnak
关于Ramanujan τ函数的两个基本问题涉及到这些数的大小和变化：(i)Ramanujan猜想：对于所有素数p。（ii）“Sato-Tate”猜想：当p→∞时，关于$$\text{d}\mu (\text{x}) = \left\{ \begin{gathered}\frac{1}{{2\pi }}\sqrt {4 - \text{x}^2 } \text{dx}\,\,\,\,\,\,\,\text{if}\,\,\left| \text{x} \right| \leqslant 2 \hfill \\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{otherwise} \hfill \\\end{gathered} \right.$$是等分布的。我们把最后一种称为半圆分布。
Two basic questions concerning the Ramanujan τ-function concern the size and variation of these numbers:(i)Ramanujan conjecture:for all primes p.(ii)“Sato-Tate” conjecture:is equidistributed with respect to $$\text{d}\mu (\text{x}) = \left\{ \begin{gathered}\frac{1}{{2\pi }}\sqrt {4 - \text{x}^2 } \text{dx}\,\,\,\,\,\,\,\text{if}\,\,\left| \text{x} \right| \leqslant 2 \hfill \\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{otherwise} \hfill \\\end{gathered} \right.$$ as p → ∞. We refer to the last as the semicircle distribution.