Liouville's summatory function L(n)

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The partial sums of Liouville's function lambda(n) give Liouville's summatory function L(n)

$L(n)\equiv \sum _{k=1}^{n}\lambda (k)=\sum _{k=1}^{n}(-1)^{\Omega (k)},\,$

where $\scriptstyle \Omega (n)\,$ is the number of primes dividing n (with multiplicity).

$\scriptstyle L(n),\,1\,\leq \,n\,\leq \,257,\,$ where $\scriptstyle L(n)\,$ is Liouville's summatory function L(n) (A002819)

16     $L(n)\,$
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margin: 0 0 0 0; z-index: 0;"></div>	<div style="position: absolute; border-style: solid; border-bottom-width: 1px; border-top-width: 1px; border-left-width: 0; border-right-width: 0; border-bottom-color: black; border-top-color: black; border-left-color: 0; border-right-color: 0; background-color: rgb(206, 223, 242); height: | 175*-7/16 | px; width: 4px; margin: 0 0 0 0; z-index: 0;"></div>	<div style="position: absolute; border-style: solid; border-bottom-width: 1px; border-top-width: 1px; border-left-width: 0; border-right-width: 0; border-bottom-color: black; border-top-color: black; border-left-color: 0; border-right-color: 0; background-color: rgb(201, 218, 237); height: | 175*-6/16 | px; width: 4px; margin: 0 0 0 0; z-index: 0;"></div>	<div style="position: absolute; border-style: solid; border-bottom-width: 1px; border-top-width: 1px; border-left-width: 0; border-right-width: 0; border-bottom-color: black; border-top-color: black; border-left-color: 0; border-right-color: 0; background-color: rgb(206, 223, 242); height: | 175*-7/16 | px; width: 4px; margin: 0 0 0 0; z-index: 0;"></div>
−16

Properties

(...)

Sequences

The partial sums $\scriptstyle L(n)\,\equiv \,\sum _{i=1}^{n}\lambda (i),\ n\,\geq \,1\,$ (Cf. A002819,) give the sequence

{1, 0, -1, 0, -1, 0, -1, -2, -1, 0, -1, -2, -3, -2, -1, 0, -1, -2, -3, -4, -3, -2, -3, -2, -1, 0, -1, -2, -3, -4, -5, -6, -5, -4, -3, -2, -3, -2, -1, 0, -1, -2, -3, -4, -5, -4, -5, -6, -5, -6, -5, -6, -7, -6, -5, ...}

See also

• A026424 Number of prime divisors (counted with multiplicity) is odd; Liouville function lambda(n) (A008836) is negative.
• A028260 Numbers n such that number of prime divisors of n (counted with multiplicity) is even; Liouville function lambda(n) (A008836) is positive.

• A056912 Odd squarefree numbers for which the number of prime divisors is odd.
• A056913 Odd squarefree numbers for which the number of prime divisors is even.

• Liouville's summatory function L(n)
• A002819 Liouville's function L(n) = partial sums of A008836.

• Liouville's function lambda(n)

• Omega(n), total number of primes dividing n