Liouville function

The Liouville lambda function, denoted by $λ(n)$ and named after Joseph Liouville, 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 primes.

Explicitly, the fundamental theorem of arithmetic states that any positive integer $n$ can be represented uniquely as a product of powers of primes: $n = p 1 a 1 \dots p k a k$ , where $p 1 < p 2 < ... < p k$ are primes and the $a j$ are positive integers. ( $1$ is given by the empty product.) The prime omega functions count the number of primes, with ( $Ω$ ) or without ( $ω$ ) multiplicity:

\omega (n)=k,

\Omega (n)=a_{1}+a_{2}+\cdots +a_{k}.

$λ(n)$ is defined by the formula

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

(sequence A008836 in the OEIS).

$λ$ is completely multiplicative since $Ω(n)$ is completely additive, i.e.: $Ω(ab) = Ω(a) + Ω(b)$ . Since $1$ has no prime factors, $Ω(1) = 0$ , so $λ(1) = 1$ .

It is related to the Möbius function $μ(n)$ . Write $n$ as $n = a 2 b$ , where $b$ is squarefree, i.e., $ω(b) = Ω(b)$ . 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{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, $λ -1 (n) = | μ (n) | = μ 2 (n)$ , the characteristic function of the squarefree integers. We also have that $λ (n) = μ 2 (n)$ .

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.

Thumb — 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.

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 $L(n)\leq 0$ 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√n for infinitely many positive integers n,^[1] while it can also be shown via the same methods that L(n) < -1.3892783√n for infinitely many positive integers n.^[2]

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.^[2]

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 $\alpha \in \mathbb {R}$ as follows for positive integers x where (as above) we have the special cases $L(x):=L_{0}(x)$ and $T(x)=L_{1}(x)$ ^[2]

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.^[2] 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.