## Mock-Gaussian behaviour

Research output: Chapter in Book/Report/Conference proceeding › Chapter

Title of host publication | Recent Perspectives in Random Matrix Theory and Number Theory |
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Date | Published - Jul 2005 |
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Pages | 337-355 |
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Number of pages | 19 |
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Publisher | Cambridge University Press |
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Volume | 322 |
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Original language | English |
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ISBN (Print) | 9780521620581 |
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Name | London Mathematical Society lecture Note Series |
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We show that the moments of the smooth counting function of a set of points encodes the same information as the $n$-level density of these points. If the points are the eigenvalues of matrices taken from the classical compact groups with Haar measure, then we show that the first few moments of the smooth counting function are Gaussian, while the distribution is not. The same phenomenon occurs for smooth counting functions of the zeros of $L$--functions, and we give examples relating to each classical compact group. The advantage of calculating moments of the counting function is that combinatorially, they are far easier to handle than the $n$-level densities.

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