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| The '''Liouville function''', denoted by λ(''n'') and named after [[Joseph Liouville]], is an important [[function (mathematics)|function]] in [[number theory]].
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| If ''n'' is a positive [[integer]], then λ(''n'') is defined as:
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| :<math>\lambda(n) = (-1)^{\Omega(n)},\,\! </math> | |
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| where [[Big Omega function|Ω(''n'')]] is the number of [[prime number|prime]] [[divisor|factors]] of ''n'', counted with multiplicity {{OEIS|A008836}}.
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| λ is [[multiplicative function|completely multiplicative]] since Ω(''n'') is completely [[additive function|additive]]. The number one has no prime factors, so Ω(1) = 0 and therefore λ(1) = 1. The Liouville function satisfies the [[Identity (mathematics)|identity]]:
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| :<math>
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| \sum_{d|n}\lambda(d) =
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| \begin{cases}
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| 1 & \text{if }n\text{ is a perfect square,} \\
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| 0 & \text{otherwise.}
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| \end{cases}
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| </math> | |
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| The Liouville function's [[Dirichlet inverse]] is the absolute value of the [[Möbius function]].
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| ==Series==
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| The [[Dirichlet series]] for the Liouville function gives the [[Riemann zeta function]] as
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| :<math>\frac{\zeta(2s)}{\zeta(s)} = \sum_{n=1}^\infty \frac{\lambda(n)}{n^s}.</math>
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| The [[Lambert series]] for the Liouville function is
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| :<math>\sum_{n=1}^\infty \frac{\lambda(n)q^n}{1-q^n} =
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| \sum_{n=1}^\infty q^{n^2} =
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| \frac{1}{2}\left(\vartheta_3(q)-1\right),</math>
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| where <math>\vartheta_3(q)</math> is the [[Jacobi theta function]].
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| ==Conjectures==
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| <div style="float: right; clear: right">
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| [[Image:Liouville.svg|thumb|none|Summatory Liouville function ''L''(''n'') up to ''n'' = 10<sup>4</sup>. The readily visible oscillations are due to the first non-trivial zero of the Riemann zeta function.]]
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| [[Image:Liouville-big.svg|thumb|none|Summatory Liouville function ''L''(''n'') up to ''n'' = 10<sup>7</sup>. Note the apparent [[scale invariance]] of the oscillations.]]
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| [[Image:Liouville-log.svg|thumb|none|Logarithmic graph of the negative of the summatory Liouville function ''L''(''n'') up to ''n'' = 2 × 10<sup>9</sup>. The green spike shows the function itself (not its negative) in the narrow region where the [[Pólya conjecture]] fails; the blue curve shows the oscillatory contribution of the first Riemann zero.]]
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| [[Image:Liouville-harmonic.svg|thumb|none|Harmonic Summatory Liouville function ''M''(''n'') up to ''n'' = 10<sup>3</sup>]]
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| </div>
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| The [[Pólya conjecture]] is a conjecture made by [[George Pólya]] in 1919. Defining
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| : <math>L(n) = \sum_{k=1}^n \lambda(k), </math>
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| the conjecture states that <math>L(n)\leq 0</math> for ''n'' > 1. This turned out to be false. The smallest counter-example is ''n'' = 906150257, found by Minoru Tanaka in 1980. It has since been shown that ''L''(''n'') > 0.0618672√''n'' for infinitely many positive integers ''n'',<ref>P. Borwein, R. Ferguson, and M. J. Mossinghoff, ''Sign Changes in Sums of the Liouville Function'', Mathematics of Computation 77 (2008), no. 263, 1681–1694.</ref> while it can also be shown that ''L''(''n'') < -1.3892783√''n'' for infinitely many positive integers ''n''.
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| Define the related sum
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| : <math>T(n) = \sum_{k=1}^n \frac{\lambda(k)}{k}.</math>
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| It was open for some time whether ''T''(''n'') ≥ 0 for sufficiently big ''n'' ≥ ''n''<sub>0</sub> (this "conjecture" is occasionally (but incorrectly) attributed to [[Pál Turán]]). This was then disproved by [[C. Brian Haselgrove|Haselgrove]] in 1958 (see the reference below), who showed that ''T''(''n'') takes negative values infinitely often. A confirmation of this positivity conjecture would have led to a proof of the [[Riemann hypothesis]], as was shown by Pál Turán.
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| ==References==
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| {{Reflist}}
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| * {{cite journal | last=Polya | first=G. | title=Verschiedene Bemerkungen zur Zahlentheorie | journal=Jahresbericht der Deutschen Mathematiker-Vereinigung | volume=28 | year=1919 | pages=31–40 }}
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| * {{cite journal|last1=Haselgrove|first1=C. B. |title=A disproof of a conjecture of Polya
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| |journal=Mathematika |volume=5|number=2 |year=1958 |pages=141–145 | doi=10.1112/S0025579300001480 | issn=0025-5793 | mr=0104638 | zbl=0085.27102 }}
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| * {{cite journal|last1=Lehman| first1=R. | title=On Liouville's function
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| |journal=Math. Comp. |volume=14 |year=1960 | pages=311–320|doi=10.1090/S0025-5718-1960-0120198-5
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| |mr=0120198}}
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| * {{cite journal|first1=M. |last1= Tanaka |title=A Numerical Investigation on Cumulative Sum of the Liouville Function | journal=Tokyo Journal of Mathematics |volume=3 |pages=187–189 |year=1980}}
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| * {{mathworld|urlname=LiouvilleFunction|title=Liouville Function}}
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| * {{springer|author=A.F. Lavrik|title=Liouville function|id=L/l059620}}
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| {{DEFAULTSORT:Liouville Function}}
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| [[Category:Multiplicative functions]]
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