Liouville 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 numbers, 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 = p₁^a₁ ⋯ p_k^a_k}}, where {{math|1=p₁ < p₂ < ... < p_k}} are primes and the {{math|1=a_j}} are positive integers. ({{math|1=1}} is given by the empty product.) The prime omega functions count the number of primes, with ({{mvar|Ω}}) or without ({{mvar|ω}}) multiplicity:

: $\omega(n) = k,$

: $\Omega(n) = a_1 + a_2 + \cdots + a_k.$

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

: $\lambda(n) = (-1)^{\Omega(n)}$

{{OEIS|A008836}}.

{{mvar|λ}} is completely multiplicative since {{math|1=Ω(n)}} is completely 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 = a²b}}, where {{mvar|b}} is squarefree, i.e., {{math|1=ω(b) = Ω(b)}}. Then

: $\lambda(n) = \mu(b).$

The sum of the Liouville function over the divisors of {{mvar|n}} is the characteristic function of the squares:

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

Möbius inversion of this formula yields

: $\lambda(n) = \sum_{d^2|n} \mu\left(\frac{n}{d^2}\right).$

The Dirichlet inverse of Liouville function is the absolute value of the Möbius function, {{math|1=λ^–1(n) = |μ(n)| = μ²(n)}}, the characteristic function of the squarefree integers.

Series

The Dirichlet series for the Liouville function is related to the Riemann zeta function by

: $\frac{\zeta(2s)}{\zeta(s)} = \sum_{n=1}^\infty \frac{\lambda(n)}{n^s}.$

Also:

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

The Lambert series for the Liouville function is

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

where $\vartheta_3(q)$ is the Jacobi theta function.

Conjectures on weighted summatory functions

Image:Liouville.svg

Image:Liouville-big.svg of the oscillations.]]

Image:Liouville-log.svg fails; the blue curve shows the oscillatory contribution of the first Riemann zero.]]

Image:Liouville-harmonic.svg

The Pólya problem is a question raised made by George Pólya in 1919. Defining

: $L(n) = \sum_{k=1}^n \lambda(k)$ {{OEIS|id=A002819}},

the problem asks whether $L(n)\leq 0$ for n > 1. The answer turns out to be no. The smallest counter-example is n = 906150257, found by Minoru Tanaka in 1980. It has since been shown that L(n) > 0.0618672{{radic|n}} for infinitely many positive integers n,{{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=1681–1694 |doi=10.1090/S0025-5718-08-02036-X |doi-access=free }} while it can also be shown via the same methods that L(n) < −1.3892783{{radic|n}} for infinitely many positive integers n.

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

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

where the $C > 0$ is some absolute limiting constant.{{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}}

Define the related sum

: $T(n) = \sum_{k=1}^n \frac{\lambda(k)}{k}.$

It was open for some time whether T(n) ≥ 0 for sufficiently big n ≥ n₀ (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 $\alpha \in \mathbb{R}$ as follows for positive integers x where (as above) we have the special cases $L(x) := L_0(x)$ and $T(x) = L_1(x)$

: $L_{\alpha}(x) := \sum_{n \leq x} \frac{\lambda(n)}{n^{\alpha}}.$

These $\alpha^{-1}$ -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 $L(x)$ precisely corresponds to the sum

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

Moreover, these functions satisfy similar bounding asymptotic relations. For example, whenever $0 \leq \alpha \leq \frac{1}{2}$ , we see that there exists an absolute constant $C_{\alpha} > 0$ such that

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

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

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

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

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

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

In particular, if we assume that the

Riemann hypothesis (RH) is true and that all of the non-trivial zeros, denoted by $\rho = \frac{1}{2} + \imath\gamma$ , of the Riemann zeta function are simple, then for any $0 \leq \alpha < \frac{1}{2}$ and $x \geq 1$ there exists an infinite sequence of $\{T_v\}_{v \geq 1}$ which satisfies that $v \leq T_v \leq v+1$ for all v such that

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

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

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

and where the remainder term

: $R_{\alpha}(x, T) \ll x^{-\alpha} + \frac{x^{1-\alpha} \log(x)}{T} + \frac{x^{1-\alpha}}{T^{1-\varepsilon} \log(x)},$

which of course tends to 0 as $T \rightarrow \infty$ . These exact analytic formula expansions again share similar properties to those corresponding to the weighted Mertens function cases. Additionally, since $\zeta(1/2) < 0$ we have another similarity in the form of $L_{\alpha}(x)$ to $M(x)$ 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

{{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

|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}}

Category:Multiplicative functions