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Liouville function : ウィキペディア英語版 | Liouville function The Liouville function, denoted by λ(''n'') and named after Joseph Liouville, is an important function in number theory. If ''n'' is a positive integer, then λ(''n'') is defined as: : where Ω(''n'') is the number of prime factors of ''n'', counted with multiplicity . λ is completely multiplicative since Ω(''n'') is completely additive, i.e.: Ω(''ab'') = Ω(''a'') + Ω(''b''). The number one has no prime factors, so Ω(1) = 0 and therefore λ(1) = 1. The Liouville function satisfies the identity: : The Liouville function's Dirichlet inverse is the absolute value of the Möbius function. ==Series== The Dirichlet series for the Liouville function is related to the Riemann zeta function by : The Lambert series for the Liouville function is : where is the Jacobi theta function.
抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)』 ■ウィキペディアで「Liouville function」の詳細全文を読む
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