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- W2324338520 abstract "Abstract Let f be an arithmetic function and S = { x 1 , … , x n } be a set of n distinct positive integers. By ( f ( x i , x j )) (resp. ( f [ x i , x j ])) we denote the n × n matrix having f evaluated at the greatest common divisor ( x i , x j ) (resp. the least common multiple [ x i , x j ]) of x , and x j as its ( i , j )-entry, respectively. The set S is said to be gcd closed if ( x i , x j ) ∈ S for 1 ≤ i , j ≤ n . In this paper, we give formulas for the determinants of the matrices ( f ( x i , x j )) and ( f [ x i , x j ]) if S consists of multiple coprime gcd-closed sets (i.e., S equals the union of S 1 , …, S k with k ≥ 1 being an integer and S 1 , …, S k being gcd-closed sets such that (lcm( S i ), lcm( S j )) = 1 for all 1 ≤ i ≠ j ≤ k ). This extends the Bourque-Ligh, Hong’s and the Hong-Loewy formulas obtained in 1993, 2002 and 2011, respectively. It also generalizes the famous Smith’s determinant." @default.
- W2324338520 created "2016-06-24" @default.
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- W2324338520 date "2016-01-01" @default.
- W2324338520 modified "2023-10-16" @default.
- W2324338520 title "Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions" @default.
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- W2324338520 doi "https://doi.org/10.1515/math-2016-0014" @default.
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