Barry,
Since I'm talking about Sato-Tate this Sunday morning at CMI, I've been trying
to understand the relationship between the Akiyama-Tanigawa conjecture
and our conjecture. First, I'll try to state each conjecture as I see it, then
I'll discuss the relationship.
The Akiyama-Tanigawa conjecture implies the GRH for elliptic curve L-functions.
It asserts that if D(K) is the L-infinity norm of the difference
between the actual
Fourier coefficient distribution and the theoretical Sato-Tate distribution
when considering *the first K primes*, then D(K) is bounded as follows.
For all eps > 0 there is a constant c such that
D(K) <= c / K^(1/2 - eps)
for all K.
Our conjecture is the following. Suppose S(M) is the L-2 norm of the difference
between the actual Fourier coefficient distribution and the theoretical
Sato-Tate distribution when considering all primes up to M. Then S(M)
is bounded as follows. There exists a number e with 1/2 <= e <= 1 such
that
S(M) <= 1/M^e
for all but finitely many M. Moreover, if the elliptic curve has rank <= 1,
then e = 1. (I'm trying to get rid of the limsup above, by the way.)
There are several differences between the two conjectures:
(1) Akiyama-Tanigawa consider the first K primes, whereas we
consider the primes up to M. Using that there are roughly
K/log(K) primes up to K, we could alternatively formulate
their statement as:
D(M) <= c / (M/log(M))^(1/2 - eps)
where D(M) is the L-infinity norm of the difference, but for the
primes up to M.
(2) L2 versus L-infinity. The main thing to note is that L2 <= L-infinity,
so
S(M) <= D(M).
So for the error term S(M) we conjecture an upper bound of
1/M^e,
for some e between 1/2 and 1, and they conjecture an upper bound of
c/(M/log(M))^(1/2 - eps)
for some c.
Taking c = 1, noting that log(M) > 1, using that S(M) <= D(M),
and that we conjecture that e >= 1/2 (based on all our data,
that is a fair conjecture), we always have
1/M^e < c/(M/log(M))^(1/2 - eps).
Thus our conjecture implies Akiyama-Tanigawa, and is in fact
stronger than it in several ways:
* L-infinity is replaced by L2
* No log factor
* 1/2-eps is replaced by a number between 1/2 and 1.
In Akiyama-Tanigawa, they prove that their conjecture implies
GRH for L(E,s). Thus our does as well.