计算二次不规则性的首次命中分析算法
First-hit analysis of algorithms for computing quadratic irregularity
AI总结：
本文利用均匀分布启发式预测计算二次域中不规则素数所需时间，并探讨收集数据验证该启发式的替代方法。
摘要：
作者先前已将正则和不规则素数的理论推广到任意全实数域。已有猜想认为，伯努利数，或等价地，黎曼ζ函数在负奇数处的值，对每个素数p在模p下是均匀分布的。这是Siegel给出的用于估计不规则素数频率的著名启发式方法的基础。迄今为止的分析表明，如果Q(√D)是一个实二次域，那么ζ函数ζ_{D}(1-2m)=ζ_{Q(√D)}(1-2m)在负奇数处的值对任何p在模p下也如预期分布。我们利用这一启发式方法来预测寻找给定数量级的二次类不规则素数所需的计算时间。我们还讨论了收集大量数据以检验该启发式方法的替代途径。
英文摘要：
The author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields. It has been conjectured that the Bernoulli numbers, or alternatively the values of the Riemann zeta function at odd negative integers, are evenly distributed modulo p for every p. This is the basis of a well-known heuristic given by Siegel for estimating the frequency of irregular primes. So far, analyses have shown that if Q(\sqrt{D}) is a real quadratic field, then the values of the zeta function ζ_{D}(1-2m)=ζ_{Q(\sqrt{D})}(1-2m) at negative odd integers are also distributed as expected modulo p for any p. We use this heuristic to predict the computational time required to find quadratic analogues of irregular primes with a given order of magnitude. We also discuss alternative ways of collecting large amounts of data to test the heuristic.
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