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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:
:\lambda(n) = (-1)^,\,\!
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:
:
\sum_\lambda(d) =
\begin
1 & \textn\text \\
0 & \text
\end

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
:\frac = \sum_^\infty \frac.
The Lambert series for the Liouville function is
:\sum_^\infty \frac =
\sum_^\infty q^ =
\frac\left(\vartheta_3(q)-1\right),
where \vartheta_3(q) 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)
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