Liouville function

In number theory, the Liouville function, named after French mathematician Joseph Liouville and denoted $\lambda (n)$, is an important arithmetic function. Its value is $1$ if $n$ is the product of an even number of prime numbers, and $-1$ if it is the product of an odd number of prime numbers.

Definition

By the fundamental theorem of arithmetic, any positive integer $n$ can be represented uniquely as a product of powers of primes:

$n=p_1^{a_1}\cdots p_k^{a_k}$,

where $p_1,\dots ,p_k$ are primes and the exponents $a_1,\dots ,a_k$ are positive integers. The prime omega function $\Omega (n)$ counts the number of primes in the factorization of $n$ with multiplicity:

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

Thus, the Liouville function is defined by

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

(sequence A008836 in the OEIS).

Properties

Since $\Omega (n)$ is completely additive; i.e., $\Omega (ab)=\Omega (a)+\Omega (b)$, then $\lambda (n)$ is completely multiplicative. Since $1$ has no prime factors, $\Omega (1)=0$, so $\lambda (1)=1$.

$\lambda (n)$ is also related to the Möbius function $\mu (n)$: if we write $n$ as $n=a^{2}b$, where $b$ is squarefree, then

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

The sum of the Liouville function over the divisors of $n$ is the characteristic function of the squares:

$\sum _{d|n}\lambda (d)=\{\begin{array}{cc}1& \text{if }n\text{ is a perfect square,}\\ 0& \text{otherwise.}\end{array}$

Möbius inversion of this formula yields

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

The Dirichlet inverse of the Liouville function is the absolute value of the Möbius function, ${\lambda }^{-1}(n)=|\mu (n)|={\mu }^2(n)$, the characteristic function of the squarefree integers.

Series

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

$\frac{\zeta (2s)}{\zeta (s)}=\sum _{n=1}^{\mathrm{\infty }}\frac{\lambda (n)}{n^s}.$

Also:

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

The Lambert series for the Liouville function is

$\sum _{n=1}^{\mathrm{\infty }}\frac{\lambda (n)q^n}{1-q^n}=\sum _{n=1}^{\mathrm{\infty }}{q}^{n^2}=\frac{1}{2}({\vartheta }_3(q)-1),$

where ${\vartheta }_3(q)$ is the Jacobi theta function.

Conjectures on weighted summatory functions

File:Liouville.svg

Summatory Liouville function L(n) up to n = 10⁴. The readily visible oscillations are due to the first non-trivial zero of the Riemann zeta function.

File:Liouville-big.svg

Summatory Liouville function L(n) up to n = 10⁷. Note the apparent scale invariance of the oscillations.

File:Liouville-log.svg

Logarithmic graph of the negative of the summatory Liouville function L(n) up to n = 2 × 10⁹. 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.

File:Liouville-harmonic.svg

Harmonic Summatory Liouville function T(n) up to n = 10³

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

$L(n)=\sum _{k=1}^n\lambda (k)$ (sequence A002819 in the OEIS),

the problem asks whether $L(n)\le 0$ for some n > 1. The answer turns out to be yes. 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, while it can also be shown via the same methods that L(n) < −1.3892783√n for infinitely many positive integers n.

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

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

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

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 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 $\alpha \in ℝ$ as follows for positive integers x where (as above) we have the special cases $L(x):=L_0(x)$ and $T(x)=L_1(x)$

$L_{\alpha }(x):=\sum _{n\le x}\frac{\lambda (n)}{n^{\alpha }}.$

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

$L(x)=\sum _{d^2\le x}M\left(\frac{x}{d^2}\right)=\sum _{d^2\le x}\sum _{n\le \frac{x}{d^2}}\mu (n).$

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

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

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

$\frac{\zeta (2\alpha +2s)}{\zeta (\alpha +s)}=s\cdot \underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{L_{\alpha }(x)}{x^{s+1}}dx,$

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

$L_{\alpha }(x)=\frac{1}{2\pi ı}\underset{{\sigma }_0-ıT}{\overset{{\sigma }_0+ıT}{\int }}\frac{\zeta (2\alpha +2s)}{\zeta (\alpha +s)}\cdot \frac{x^s}{s}ds+E_{\alpha }(x)+R_{\alpha }(x,T),$

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 $R_{\alpha }(x,T)\to 0$ as $T\to \mathrm{\infty }$.

In particular, if we assume that the Riemann hypothesis (RH) is true and that all of the non-trivial zeros, denoted by $\rho =\frac{1}{2}+ı\gamma$, of the Riemann zeta function are simple, then for any $0\le \alpha <\frac{1}{2}$ and $x\ge 1$ there exists an infinite sequence of $\{T_v{\}}_{v\ge 1}$ which satisfies that $v\le T_v\le v+1$ 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 $0<\epsilon <\frac{1}{2}-\alpha$ we define

$I_{\alpha }(x):=\frac{1}{2\pi ı\cdot x^{\alpha }}\underset{\epsilon +\alpha -ı\mathrm{\infty }}{\overset{\epsilon +\alpha +ı\mathrm{\infty }}{\int }}\frac{\zeta (2s)}{\zeta (s)}\cdot \frac{x^s}{(s-\alpha )}ds,$

and where the remainder term

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

which of course tends to 0 as $T\to \mathrm{\infty }$. 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 $M(x)$ insomuch 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.