Connection to zeroes on the critical line

•Suppose (for fictitious expositional simplicity) n = 0 for a cusp form
on SL(2,Z). It is not difficult to handle arbitrary n.

•Let H(t) = Mt(1/2+it). Then
H(t) = H(-t) is real.

•Let y1/T be an approximate
identity such that My(1/2+it) ¸ 0.

•

•If L(s) has only a finite number of zeroes on the
critical line, then the following
integral must also be of order T:

•But it cannot if t(x) vanishes to infinite order at x=1 (y is concentrated near a point where t behaves as if it is zero).

•In that case this integral decays as O(T-N) for any N > 0!

•The above was for a cusp form on SL(2,Z). For congruence groups, the point x=1 changes to pq, q = level. The bound S(T,x) = Oe(T1/2+e)
shows that the last integral
is still o(T) with room to spare.

•New phenomena: numerically that integral decays only like T1/2 for q=11.