Abstract
In this paper we apply to the zeros of families of Lfunctions with orthogonal or symplectic symmetry the method that Conrey and Snaith [27] used to calculate the
ncorrelation of
the zeros of the Riemann zeta function. This method uses the Ratios Conjectures [21] for averages
of ratios of zeta or
Lfunctions. Katz and Sarnak [57] conjecture that the zero statistics of families
of
Lfunctions have an underlying symmetry relating to one of the classical compact groups
U(N),
O(N) and
USp(2N). Here we complete the work already done with
U(N) [27] to show how new
methods for calculating the
nlevel densities of eigenangles of random orthogonal or symplectic matrices can be used to create explicit conjectures for the
nlevel densities of zeros of
Lfunctions with
orthogonal or symplectic symmetry, including all the lower order terms. We show how the method
used here results in formulae that are easily modied when the test function used has a restricted
range of support, and this will facilitate comparison with rigorous number theoretic
nlevel density
results.
Original language  English 

Pages (fromto)  193 
Number of pages  93 
Journal  Memoirs of the American Mathematical Society 
Volume  251 
Issue number  1194 
Early online date  11 Sep 2017 
DOIs  
Publication status  Published  30 Dec 2017 
Keywords
 1M50
 15B52
 11M26
 11G05
 11M06
 15B10
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Professor Nina C Snaith
 School of Mathematics  Professor of Mathematical Physics
 Applied Mathematics
 Mathematical Physics
 Pure Mathematics
Person: Academic , Member