Divisor Problems and the Pair Correlation for the Fractional Parts of n2α
Divisor Problems and the Pair Correlation for the Fractional Parts of n2α
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n2α 小数部分的除数问题和配对相关性
DOI：
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发表时间：
2010
期刊：
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通讯作者：
J. L. Truelsen
中科院分区：
文献类型：
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作者：
J. L. Truelsen
Z. Rudnick和P. Sarnak证明了n2 α的小数部分的对相关对于几乎所有α都是泊松的。然而，他们没能找到一个特定的α。我们证明了这个问题与确定(a, b, r)∈n3的个数的问题有关，使得a≤M， b≤N, r≤K，并且pab≡r(q)对于p和q的素数。通过对K， M， N和q的相对大小的适当假设，人们应该期望有KMN/q这样的三元组渐近，我们将证明这在平均情况下成立。
Z. Rudnick and P. Sarnak have proved that the pair correlation for the fractional parts of n 2 α is Poissonian for almost all α. However, they were not able to find a specific α for which it holds. We show that the problem is related to the problem of determining the number of (a, b, r) ∈ N 3 such that a ≤ M, b ≤ N, r ≤ K, and pab ≡ r(q) for p and q coprime. With suitable assumptions on the relative size of K, M, N, and q, one should expect there to be KMN/q such triples asymptotically and we will show that this holds on average.