0
$\begingroup$

I would like to know if, under Ramanujan conjecture, the following three distributions are known or conjectured to match:

  1. the distribution of spacings between Satake parameters of an L-function $F$ at all unramified primes on the unit circle
  2. the distribution of the spacings between non trivial zeros of $F$ under the analogue of RH for $F$ on a circle of the Riemann sphere whose stereographic projection is the critical line
  3. the spacings of eigenvalues of some random matrix along the lines of Katz-Sarnak philosophy.
Improve this question
$\endgroup$

1 Answer 1

Reset to default
3
$\begingroup$

If the first item refers to the set of Satake parameters for $X \le p <2 X$ for large $X$, then that distribution will not match the others. This is because the Satake parameters for different primes do not "see" each other: they will not have (quadratic) repulsion as is expected or known in the other two cases.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .