Lower bounds for the largest eigenvalue of the gcd matrix on {1, 2,..., n}

被引：1

作者：

Merikoski, Jorma K. ^{[1
]}

机构：

[1] Univ Tampere, Sch Informat Sci, Kanslerinrinne 1, FI-33014 Tampere, Finland

来源：

CZECHOSLOVAK MATHEMATICAL JOURNAL | 2016年 / 66卷 / 03期

关键词：

eigenvalue bounds; greatest common divisor matrix; COMMON DIVISOR MATRICES;

D O I：

10.1007/s10587-016-0307-5

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

Consider the nxn matrix with (i, j)'th entry gcd (i, j). Its largest eigenvalue lambda (n) and sum of entries s (n) satisfy lambda (n) > s (n) /n. Because s (n) cannot be expressed algebraically as a function of n, we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S.Hong, R.Loewy (2004). We also conjecture that lambda (n) > 6 pi(-2) nlogn for all n. If n is large enough, this follows from F.Balatoni (1969).

引用

页码：1027 / 1038

页数：12

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