We study the 1-level density of low-lying zeros of Dirichlet
-functions in the family
of all characters modulo
,
with
.
For test functions whose Fourier transform is supported in
, we
calculate this quantity beyond the square root cancellation expansion arising from the
-function
ratios conjecture of Conrey, Farmer and Zirnbauer. We discover the existence of a
new lower-order term which is not predicted by this powerful conjecture. This is the
first family where the 1-level density is determined well enough to see a term which is
not predicted by the ratios conjecture, and proves that the exponent of the error
term
in the ratios conjecture is best possible. We also give more precise results when the
support of the Fourier transform of the test function is restricted to the interval
. Finally
we show how natural conjectures on the distribution of primes in arithmetic progressions
allow one to extend the support. The most powerful conjecture is Montgomery’s, which
implies that the ratios conjecture’s prediction holds for any finite support up to an
error
.
Keywords
low-lying zeros, Dirichlet L-functions, ratios conjecture,
primes in arithmetic progressions, random matrix theory