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.