Delta(E,X) <= 1/X^(1 - c(E)/log(X))
What is a
conjecturally optimal choice?
The function E |--> c(E) is very likely *not* a function only of the
rank of E. It is probably a function of the (low lying) zeros of
L(E,s) on the critical line. E.g., curves with no low zeros have a
small c(E); curves with low-lying zeros (or actual zeros) have a
bigger c(E).
Some examples of graphs of Delta(E_r, X) and 1/X^(1 - s/log(X))
for various values of s and elliptic curves of given rank.
Guess:
If E is any elliptic curve over QQ, then
Delta(E, X) <= 1 / X^(1 - (1/2 + rank(E))/log(X))
for *all* X.
This is *FALSE* if (1/2 + rank(E)) is replaced by rank(E) -- For the
rank 2 curve [0,1,0,-125,-424], the inequality fails for a while
around X = 6000.
Give lots of evidence with a wide range of curves of each rank.
It is ** FALSE ** -- This rank 0 curve (with trivial torsion)
is a counterexample: [1,1,0,-6382494330,-196446593263212]