Editing Liouville function

{{Short description|Arithmetic function}}
The '''Liouville lambda function''', denoted by {{math|1=λ(''n'')}} and named after [[Joseph Liouville]], is an important [[arithmetic function]]. 
Its value is {{math|1=+1}} if {{mvar|n}} is the product of an even number of [[prime number]]s, and {{math|1=−1}} if it is the product of an odd number of primes. 

Explicitly, the [[fundamental theorem of arithmetic]] states that any positive [[integer]] {{mvar|n}} can be represented uniquely as a product of powers of primes: {{math|1=''n'' = ''p''<sub>1</sub><sup>''a''<sub>1</sub></sup> &ctdot; ''p''<sub>''k''</sub><sup>''a''<sub>''k''</sub></sup>}}, where  {{math|1=''p''<sub>1</sub> < ''p''<sub>2</sub> < ... < ''p''<sub>''k''</sub>}} are primes and the {{math|1=''a<sub>j</sub>''}} are positive integers. ({{math|1=1}} is given by the empty product.) The [[prime omega function]]s  count the number of primes,  with ({{mvar|Ω}}) or without ({{mvar|ω}}) multiplicity:

: <math> \omega(n) = k, </math>
: <math> \Omega(n) = a_1 + a_2 + \cdots + a_k. </math>

{{math|1=λ(''n'')}} is defined by the formula

: <math> \lambda(n) = (-1)^{\Omega(n)} </math>

{{OEIS|A008836}}.

{{mvar|λ}} is [[multiplicative function|completely multiplicative]] since {{math|1=Ω(''n'')}} is completely [[additive function|additive]], i.e.: {{math|1=Ω(''ab'') = Ω(''a'') + Ω(''b'')}}. Since {{math|1}} has no prime factors, {{math|1=Ω(1) = 0}}, so {{math|1=λ(1) = 1}}.

It is related to the [[Möbius function]] {{math|1=μ(''n'')}}. Write {{mvar|n}} as {{math|1=''n'' = ''a''<sup>2</sup>''b''}}, where {{mvar|b}} is [[squarefree]], i.e., {{math|1=ω(''b'') = Ω(''b'')}}.  Then

: <math> \lambda(n) = \mu(b). </math>

The sum of the Liouville function over the [[divisor]]s of {{mvar|n}} is the [[indicator function|characteristic function]] of the [[square (algebra)|squares]]:

:<math>
\sum_{d|n}\lambda(d) =
\begin{cases}
1 & \text{if }n\text{ is a perfect square,} \\
0 & \text{otherwise.}
\end{cases}
</math>

[[Möbius inversion formula|Möbius inversion]] of this formula yields
:<math>\lambda(n) = \sum_{d^2|n} \mu\left(\frac{n}{d^2}\right).</math>

The [[Dirichlet inverse]] of Liouville function is the absolute value of the Möbius function, {{math|1=&lambda;<sup>–1</sup>(''n'') = &vert;&mu;(''n'')&vert; = &mu;<sup>2</sup>(''n'')}}, the characteristic function of the squarefree integers.

==Series==
The [[Dirichlet series]] for the Liouville function is related to the [[Riemann zeta function]] by

:<math>\frac{\zeta(2s)}{\zeta(s)} = \sum_{n=1}^\infty \frac{\lambda(n)}{n^s}.</math>

Also:

:<math>\sum\limits_{n=1}^{\infty} \frac{\lambda(n)\ln n}{n}=-\zeta(2)=-\frac{\pi^2}{6}.</math>

The [[Lambert series]] for the Liouville function is

:<math>\sum_{n=1}^\infty \frac{\lambda(n)q^n}{1-q^n} = 
\sum_{n=1}^\infty q^{n^2} = 
\frac{1}{2}\left(\vartheta_3(q)-1\right),</math>

where <math>\vartheta_3(q)</math> is the [[Jacobi theta function]].

==Conjectures on weighted summatory functions==
<div style="float: right; clear: right">
[[Image:Liouville.svg|thumb|none|Summatory Liouville function ''L''(''n'') up to ''n''&nbsp;=&nbsp;10<sup>4</sup>. The readily visible oscillations are due to the first non-trivial zero of the Riemann zeta function.]]
[[Image:Liouville-big.svg|thumb|none|Summatory Liouville function ''L''(''n'') up to ''n''&nbsp;=&nbsp;10<sup>7</sup>. Note the apparent [[scale invariance]] of the oscillations.]]
[[Image:Liouville-log.svg|thumb|none|Logarithmic graph of the negative of the summatory Liouville function ''L''(''n'') up to ''n''&nbsp;=&nbsp;2&nbsp;×&nbsp;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.]]
[[Image:Liouville-harmonic.svg|thumb|none|Harmonic Summatory Liouville function ''T''(''n'') up to ''n''&nbsp;=&nbsp;10<sup>3</sup>]]
</div>

The [[Pólya conjecture|Pólya problem]] is a question raised made by [[George Pólya]] in 1919. Defining

: <math>L(n) = \sum_{k=1}^n \lambda(k)</math> {{OEIS|id=A002819}},

the problem asks whether <math>L(n)\leq 0</math> for ''n''&nbsp;>&nbsp;1. The answer turns out to be no.  The smallest counter-example is ''n''&nbsp;=&nbsp;906150257, found by Minoru Tanaka in 1980. It has since been shown that ''L''(''n'')&nbsp;>&nbsp;0.0618672{{radic|''n''}} for infinitely many positive integers ''n'',<ref>{{cite journal |first=P. |last=Borwein |first2=R. |last2=Ferguson |first3=M. J. |last3=Mossinghoff |title=Sign Changes in Sums of the Liouville Function |journal=Mathematics of Computation |volume=77 |year=2008 |issue=263 |pages=1681&ndash;1694 |doi=10.1090/S0025-5718-08-02036-X |doi-access=free }}</ref> while it can also be shown via the same methods that ''L''(''n'')&nbsp;<&nbsp;−1.3892783{{radic|''n''}} for infinitely many positive integers ''n''.<ref name="HUMPHRIES-WEIGHTED-SUMFUNCS" />

For any <math>\varepsilon > 0</math>, assuming the Riemann hypothesis, we have that the summatory function <math>L(x) \equiv L_0(x)</math> is bounded by 

:<math>L(x) = O\left(\sqrt{x} \exp\left(C \cdot \log^{1/2}(x) \left(\log\log x\right)^{5/2+\varepsilon}\right)\right),</math> 

where the <math>C > 0</math> is some absolute limiting constant.<ref name="HUMPHRIES-WEIGHTED-SUMFUNCS">{{cite journal |arxiv = 1108.1524|doi = 10.1016/j.jnt.2012.08.011|doi-access=free|title = The distribution of weighted sums of the Liouville function and Pólyaʼs conjecture|journal = Journal of Number Theory|volume = 133|issue = 2|pages = 545–582|year = 2013|last1 = Humphries|first1 = Peter}}</ref>

Define the related sum

: <math>T(n) = \sum_{k=1}^n \frac{\lambda(k)}{k}.</math>

It was open for some time whether ''T''(''n'')&nbsp;≥&nbsp;0 for sufficiently big ''n'' ≥ ''n''<sub>0</sub> (this conjecture is occasionally–though incorrectly–attributed to [[Pál Turán]]).  This was then disproved by {{harvtxt|Haselgrove|1958}}, 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]]. 

===Generalizations===

More generally, we can consider the weighted summatory functions over the Liouville function defined for any <math>\alpha \in \mathbb{R}</math> as follows for positive integers ''x'' where (as above) we have the special cases <math>L(x) := L_0(x)</math> and <math>T(x) = L_1(x)</math> <ref name="HUMPHRIES-WEIGHTED-SUMFUNCS"/>

:<math>L_{\alpha}(x) := \sum_{n \leq x} \frac{\lambda(n)}{n^{\alpha}}.</math> 

These <math>\alpha^{-1}</math>-weighted summatory functions are related to the [[Mertens function]], or weighted summatory functions of the [[Moebius function]]. In fact, we have that the so-termed non-weighted, or ordinary function <math>L(x)</math> precisely corresponds to the sum 

:<math>L(x) = \sum_{d^2 \leq x} M\left(\frac{x}{d^2}\right) = \sum_{d^2 \leq x} \sum_{n \leq \frac{x}{d^2}} \mu(n).</math> 

Moreover, these functions satisfy similar bounding asymptotic relations.<ref name="HUMPHRIES-WEIGHTED-SUMFUNCS"/> For example, whenever <math>0 \leq \alpha \leq \frac{1}{2}</math>, we see that there exists an absolute constant <math>C_{\alpha} > 0</math> such that 

:<math>L_{\alpha}(x) = O\left(x^{1-\alpha}\exp\left(-C_{\alpha} \frac{(\log x)^{3/5}}{(\log\log x)^{1/5}}\right)\right).</math>

By an application of [[Perron's formula]], or equivalently by a key (inverse) [[Mellin transform]], we have that 

:<math>\frac{\zeta(2\alpha+2s)}{\zeta(\alpha+s)} = s \cdot \int_1^{\infty} \frac{L_{\alpha}(x)}{x^{s+1}} dx,</math> 

which then can be inverted via the [[Mellin transform|inverse transform]] to show that for <math>x > 1</math>, <math>T \geq 1</math> and <math>0 \leq \alpha < \frac{1}{2}</math> 

:<math>L_{\alpha}(x) = \frac{1}{2\pi\imath} \int_{\sigma_0-\imath T}^{\sigma_0+\imath T} \frac{\zeta(2\alpha+2s)}{\zeta(\alpha+s)} 
     \cdot \frac{x^s}{s} ds + E_{\alpha}(x) + R_{\alpha}(x, T), </math>

where we can take <math>\sigma_0 := 1-\alpha+1 / \log(x)</math>, and with the remainder terms defined such that <math>E_{\alpha}(x) = O(x^{-\alpha})</math> and <math>R_{\alpha}(x, T) \rightarrow 0</math> as <math>T \rightarrow \infty</math>. 

In particular, if we assume that the 
[[Riemann hypothesis]] (RH) is true and that all of the non-trivial zeros, denoted by <math>\rho = \frac{1}{2} + \imath\gamma</math>, of the [[Riemann zeta function]] are [[simple zero|simple]], then for any <math>0 \leq \alpha < \frac{1}{2}</math> and <math> x \geq 1</math> there exists an infinite sequence of <math>\{T_v\}_{v \geq 1}</math> which satisfies that <math>v \leq T_v \leq v+1</math> for all ''v'' such that 

:<math>L_{\alpha}(x) = \frac{x^{1/2-\alpha}}{(1-2\alpha) \zeta(1/2)} + \sum_{|\gamma| < T_v} \frac{\zeta(2\rho)}{\zeta^{\prime}(\rho)} \cdot 
     \frac{x^{\rho-\alpha}}{(\rho-\alpha)} + E_{\alpha}(x) + R_{\alpha}(x, T_v) + I_{\alpha}(x), </math> 

where for any increasingly small <math>0 < \varepsilon < \frac{1}{2}-\alpha</math> we define 

:<math>I_{\alpha}(x) := \frac{1}{2\pi\imath \cdot x^{\alpha}} \int_{\varepsilon+\alpha-\imath\infty}^{\varepsilon+\alpha+\imath\infty} 
     \frac{\zeta(2s)}{\zeta(s)} \cdot \frac{x^s}{(s-\alpha)} ds,</math> 

and where the remainder term 

:<math>R_{\alpha}(x, T) \ll x^{-\alpha} + \frac{x^{1-\alpha} \log(x)}{T} + \frac{x^{1-\alpha}}{T^{1-\varepsilon} \log(x)}, </math> 

which of course tends to ''0'' as <math>T \rightarrow \infty</math>. These exact analytic formula expansions again share similar properties to those corresponding to the weighted [[Mertens function]] cases. Additionally, since <math>\zeta(1/2) < 0</math> we have another similarity in the form of <math>L_{\alpha}(x)</math> to <math>M(x)</math> in so much as the dominant leading term in the previous formulas predicts a negative bias in the values of these functions over the positive natural numbers ''x''.

==References==
{{Reflist}}
* {{cite journal | last=Pólya | first=G. | title=Verschiedene Bemerkungen zur Zahlentheorie | journal=Jahresbericht der Deutschen Mathematiker-Vereinigung | volume=28 | year=1919 | pages=31–40 }}
* {{cite journal|last1=Haselgrove|first1=C. Brian | author-link=C. Brian Haselgrove | title=A disproof of a conjecture of Pólya
|journal=[[Mathematika]] |volume=5|number=2 |year=1958 |pages=141–145 | doi=10.1112/S0025579300001480 | issn=0025-5793 | mr=0104638 | zbl=0085.27102 }}
* {{cite journal|last1=Lehman| first1=R. | title=On Liouville's function
|journal=Mathematics of Computation |volume=14 | issue=72 |year=1960 | pages=311&ndash;320|doi=10.1090/S0025-5718-1960-0120198-5
|mr=0120198|doi-access=free}}
* {{cite journal|first1=Minoru|last1= Tanaka |title=A Numerical Investigation on Cumulative Sum of the Liouville Function | journal=[[Tokyo Journal of Mathematics]] |volume=3 | issue=1 |pages=187–189 |year=1980 | mr=0584557 | doi=10.3836/tjm/1270216093|doi-access=free }}
* {{mathworld|urlname=LiouvilleFunction|title=Liouville Function}}
* {{springer|author=A.F. Lavrik|title=Liouville function|id=L/l059620}}

{{DEFAULTSORT:Liouville Function}}
[[Category:Multiplicative functions]]