ASYMPTOTIC BEHAVIOR OF LARGEST EIGENVALUE OF MATRICES ASSOCIATED WITH COMPLETELY EVEN FUNCTIONS (MOD r)
ASYMPTOTIC BEHAVIOR OF LARGEST EIGENVALUE OF MATRICES ASSOCIATED WITH COMPLETELY EVEN FUNCTIONS (MOD r)
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DOI：
10.1142/s1793557108000217
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发表时间：
2008-06
影响因子：
0.8
通讯作者：
Shaofang Hong
中科院分区：
文献类型：
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作者：
Shaofang Hong
给出一个任意严格递增的正整数序列${xi}^{inty}_{i=1}$，设S_n={x_1，…设q≥1是给定的整数，f是算术函数。设$lambda^{(1)}_{n}\leq\ldots^{(N)}_n}$是矩阵(f(xi，xj))的特征值，其中f在xi和xj的最大公约数(xi，xj)处求值为其i，j-项。对于任意素数p和任意整数m≥1和(f*−)(D)(D)和(f*≥)(D)和(f*μ)(D)>；0，当d|x对任意$x在{x_n}^{\inty}_{n=1}$时，$\lambda^{(n-q+1)}_{n}$趋于无穷大时，得到了仅依赖于x1和n的下界。
Given an arbitrary strictly increasing infinite sequence $\{x_i\}^{\infty}_{i=1}$ of positive integers, let Sn = {x1,…, xn} for any integer n ≥ 1. Let q ≥ 1 be a given integer and f an arithmetical function. Let $\lambda^{(1)}_{n} \leq \ldots \leq \lambda^{(n)}_{n}$ be the eigenvalues of the matrix (f(xi, xj)) having f evaluated at the greatest common divisor (xi, xj) of xi and xj as its i, j-entry. We obtain a lower bound depending only on x1 and n for $\lambda^{(1)}_{n}$ if (f * μ)(d) 1 and f(pm) ≥ f(2)f(pm−1) for any prime p and any integer m ≥ 1 and (f *μ)(d) > 0 whenever d|x for any $x \in \{x_n\}^{\infty}_{n=1}$, then $\lambda^{(n-q+1)}_{n}$ approaches infinity when n goes to infinity.