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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:
 
:<math>\lambda(n) = (-1)^{\Omega(n)},\,\! </math>
 
where [[Big Omega function|&Omega;(''n'')]] is the number of [[prime number|prime]] [[divisor|factors]] of ''n'', counted with multiplicity {{OEIS|A008836}}.
 
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]]:
 
:<math>
\sum_{d|n}\lambda(d) =
\begin{cases}
1 & \text{if }n\text{ is a perfect square,} \\
0 & \text{otherwise.}
\end{cases}
</math>
 
The Liouville function's [[Dirichlet inverse]] is the absolute value of the [[Möbius function]].
 
==Series==
The [[Dirichlet series]] for the Liouville function gives the [[Riemann zeta function]] as
 
:<math>\frac{\zeta(2s)}{\zeta(s)} = \sum_{n=1}^\infty \frac{\lambda(n)}{n^s}.</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==
<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 ''M''(''n'') up to ''n''&nbsp;=&nbsp;10<sup>3</sup>]]
</div>
 
The [[Pólya conjecture]] is a conjecture made by [[George Pólya]] in 1919. Defining
 
: <math>L(n) = \sum_{k=1}^n \lambda(k), </math>
 
the conjecture states that <math>L(n)\leq 0</math> for ''n''&nbsp;>&nbsp;1. This turned out to be false.  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√''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&ndash;1694.</ref> while it can also be shown that ''L''(''n'')&nbsp;<&nbsp;-1.3892783√''n'' for infinitely many positive integers ''n''.
 
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 (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.
 
==References==
{{Reflist}}
* {{cite journal | last=Polya | 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. B. |title=A disproof of a conjecture of Polya
|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=Math. Comp. |volume=14 |year=1960 | pages=311&ndash;320|doi=10.1090/S0025-5718-1960-0120198-5
|mr=0120198}}
* {{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&ndash;189 |year=1980}}
* {{mathworld|urlname=LiouvilleFunction|title=Liouville Function}}
* {{springer|author=A.F. Lavrik|title=Liouville function|id=L/l059620}}
 
{{DEFAULTSORT:Liouville Function}}
[[Category:Multiplicative functions]]

Revision as of 20:54, 14 February 2014

The most important thing is that you should be aware of all the causes so that you can prevent this ailment from occurring. If you are prone to yeast infections, natural cures are definitely worth looking into as it is likely they will provide the results you need without any of the side-effects possible from taking medication to cure yeast infection. When the amount of Candida increase, it leads to imbalance and causes yeast infection, leading to uncomfortable and embarrassing feeling for its sufferers. If you want to know a lot more about this treatment, you really should study on. If left untended, they can become acute and severe which may actually become life-threatening.

If untreated it can lead to other infections deep inside the woman's female parts and also can be transmitted to their partner during sex. DIM-Plus -- This is a great product that I have used myself. The word bacteriophage comes from the Greek "phagein" which means eater of bacteria. But as with all ailments, always consult your doctor for a proper diagnosis and an appropriate treatment program to get rid of your yeast infection. We all have various types of bacteria in our bodies and most of them are needed for all round good health.

This impacts the way in which which you deal with your infection simply because you not just need to get relief out of your signs and symptoms, you also need to do the function that your immune method should have performed and obtain the Candida back beneath manage. Fill your tub and add one cup of oatmeal engrossed. Wear panties made of breathable materials such as cotton or at least panties with cotton crotch (preferably white). While there could be various other reasons of penis yeast infection, the most common one is Candida bacteria. But the important thing here is that it has to have the live cultures.

- Echinacea and Golden seal are herbs which help to boost the immune system and prevent infection. A little reasearch will help you find out what the best solution is for you. Typical of these herbs are licorice, chamomile, goldenseal and Pau-d'arco. By ensuring there is enough air around the vagina, you can keep the accumulation of the fungus at bay. ” These products are lining the shelves of supermarkets and pharmacies everywhere, and advertisements for them have infiltrated magazines, radio ads, and TV commercials.

In the event your medical professional believes that a health problem, such as diabetic issues , may well be bringing about a yeast infection, you could possibly additionally be tested with regard to that ailment. Garlic is another option you have at your disposal. This typically happens only during times of menstruation. The yeast infection symptom a person gets depends on where the infection is located and on how severe the infection is. Of course, you can use probiotics capsules and suppositories from the pharmacy as an alternative.

In case you loved this short article and you wish to receive much more information about how to cure yeast infection naturally kindly visit our web-site.

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