A p-adic Study of the Partial Sums of the Harmonic Series
A p-adic Study of the Partial Sums of the Harmonic Series
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DOI：
10.1080/10586458.1994.10504298
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发表时间：
1994
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影响因子：
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通讯作者：
D. Boyd
中科院分区：
文献类型：
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作者：
D. Boyd
设Hn=1+1/2+…+是调和级数的第n次部分和。Woltenholme的一个经典结果表明，如果p>；3是素数，则Hp-L的分子可被p2整除。这里我们考虑，对于给定的素数p，n的集合Jp，其中p除以Hn的分子。这个集合Jp以前被确定为p=2，3，5，7。我们的结果之一是J11正好包含638个整数，其中最大的是31位十进制数字。我们确定除83、127和397之外，所有的p<；550指数都是Jp。这个计算是基于一个新的p-p收敛公式Hpn-Hn/p。我们描述了一个基于分支过程的集合Jp的概率模型。该模型预测了|Jp|=O(p2(Loglogp)2+∊)，并且有无穷多个p满足|Jp|≥p2(Loglogp)2。这加强了Eswarathasan和Levine早先的猜想，即对所有p来说|Jp|是有限的。该模型的另一个预测是，对于w，将存在无穷多个对(n，p)。
Let H n = 1 + ½ + … + be the n-th partial sum of the harmonic series. A classical result of Wolstenholme states that, if p > 3 is prime, the numerator of H p –l is divisible by p 2. Here we consider, for a given prime p, the set J p of n for which p divides the numerator of H n . This set J p had been previously determined for p = 2,3,5,7. One of our results is that J 11 contains exactly 638 integers, the largestof which is a number of 31 decimal digits. We determine J p for all p < 550 with three exceptions: 83, 127 and 397. The computation is based on a new p-adically convergent formula for the quantity H pn – H n /p. We describe a probabilistic model for the sets J p , based on branching processes. The model predicts that |J p | = O(p 2(log log p)2+∊), and that there are infinitely many p with |J p | ≥ p 2(log log p)2. This strengthens an earlier conjecture of Eswarathasan and Levine that |J p| is finite for all p. Another prediction of the model is that there will be infin itely many pairs (n,p) for w...