The moments of the coefficients of elliptic curve L-functions are related to numerous arithmetic problems. Rosen and Silverman proved a conjecture of Nagao relating the first moment of one-parameter families satisfying Tate's conjecture to the rank of the corresponding elliptic surface over Q(T); one can also construct families of moderate rank by finding families with large first moments. Michel proved that if j(T) is not constant, then the second moment of the family is of size p^2 + O(p^(3/2)); these two moments show that for suitably small support the behavior of zeros near the central point agree with that of eigenvalues from random matrix ensembles, with the higher moments impacting the rate of convergence. In his thesis, Miller noticed a negative bias in the second moment of every one-parameter family of elliptic curves over the rationals whose second moment had a calculable closed-form expression, specifically the first lower order term which does not average to zero is on average negative. This Bias Conjecture is confirmed for many families; however, these are highly non-generic families whose resulting Legendre sums can be determined. Inspired by the recent successes by Yang-Hui He, Kyu-Hwan Lee, Thomas Oliver, Alexey Pozdnyakov and others in investigations of murmurations of elliptic curve coefficients with machine learning techniques, we pose a similar problem for trying to understand the Bias Conjecture. As a start to this program, we numerically investigate the Bias Conjecture for a family whose bias is positive for half the primes. Since the numerics do not offer conclusive evidence that negative bias for the other half is enough to overwhelm the positive bias, the Bias Conjecture cannot be verified for the family.

翻译：椭圆曲线L函数系数的矩与众多算术问题相关。Rosen和Silverman证明了Nagao猜想，该猜想将满足Tate猜想的单参数族的第一矩与Q(T)上对应椭圆曲面的秩联系起来；通过寻找具有较大第一矩的族，也可以构造中等秩的族。Michel证明了若j(T)非常数，则该族的第二矩大小为p^2 + O(p^(3/2))；这两个矩表明，对于适当小的支撑集，中央点附近的零点行为与随机矩阵系综的特征值一致，而高阶矩则影响收敛速度。在其学位论文中，Miller注意到所有有理数域上椭圆曲线单参数族的第二矩（其可计算闭式表达式）存在负偏差，具体而言，第一个非零平均的低阶项平均值为负。该偏差猜想已在许多族中得到验证，但这些族均为高度非典型的族，其Legendre和可被确定。受Yang-Hui He、Kyu-Hwan Lee、Thomas Oliver、Alexey Pozdnyakov等人在利用机器学习技术研究椭圆曲线系数群聚现象方面最新成功的启发，我们对尝试理解偏差猜想提出了类似的问题。作为该计划的起点，我们通过数值方法研究了一个族（其半数素数偏差为正）的偏差猜想。由于数值结果并未提供充分证据表明另一半素数的负偏差足以压倒正偏差，因此无法验证该族的偏差猜想。