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:   $n = p_1^\cdots p_k^$   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:

:''ω''(''n'') = ''k'',
:Ω(''n'') = ''a''₁ + ''a''₂ + ... + ''a''_''k''.

λ(''n'') is defined by the formula

:$\lambda(n) = (-1)^.$ .

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:

:$\sum_\lambda(d) = \begin 1 & \textn\text \\ 0 & \text \end$

Möbius inversion of this formula yields
:$\lambda(n) = \sum_ \mu\left(\frac\right).$

The Dirichlet inverse of Liouville function is the absolute value of the Möbius function, $\lambda^(n) = , \mu(n), = \mu^2(n),$ the characteristic function of the squarefree integers. We also have that $\lambda(n) \mu(n) = \mu^2(n)$.

Series

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

:$\frac = \sum_^\infty \frac.$

Also:

:$\sum\limits_^ \frac=-\zeta(2)=-\frac.$

The Lambert series for the Liouville function is

:$\sum_^\infty \frac = \sum_^\infty q^ = \frac\left(\vartheta_3(q)-1\right),$

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

Conjectures on weighted summatory functions

The Pólya conjecture is a conjecture made by George Pólya in 1919. Defining

: $L(n) = \sum_^n \lambda(k)$ ,

the conjecture states that $L(n)\leq 0$ for ''n'' > 1. This turned out to be false. The smallest counter-example is ''n'' = 906150257, found by Minoru Tanaka in 1980. It has since been shown that ''L''(''n'') > 0.0618672 for infinitely many positive integers ''n'', while it can also be shown via the same methods that ''L''(''n'') < -1.3892783 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

:$L(x) = O\left(\sqrt \exp\left(C \cdot \log^(x) \left(\log\log x\right)^\right)\right),$

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

Define the related sum

: $T(n) = \sum_^n \frac.$

It was open for some time whether ''T''(''n'') ≥ 0 for sufficiently big ''n'' ≥ ''n''₀ (this conjecture is occasionally–though incorrectly–attributed to Pál Turán). This was then disproved by , 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 $\alpha \in \mathbb$ as follows for positive integers ''x'' where (as above) we have the special cases $L(x) := L_0(x)$ and $T(x) = L_1(x)$

:$L_(x) := \sum_ \frac.$

These $\alpha^$-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

:$L(x) = \sum_ M\left(\frac\right) = \sum_ \sum_ \mu(n).$

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

:$L_(x) = O\left(x^\exp\left(-C_ \frac\right)\right).$

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

:$\frac = s \cdot \int_1^ \frac dx,$

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

:$L_(x) = \frac \int_^ \frac \cdot \frac ds + E_(x) + R_(x, T),$

where we can take $\sigma_0 := 1-\alpha+1 / \log(x)$, and with the remainder terms defined such that $E_(x) = O(x^)$ and $R_(x, T) \rightarrow 0$ 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 $\rho = \frac + \imath\gamma$, of the Riemann zeta function are simple, then for any $0 \leq \alpha < \frac$ and $x \geq 1$ there exists an infinite sequence of $\_$ which satisfies that $v \leq T_v \leq v+1$ for all ''v'' such that

:$L_(x) = \frac + \sum_ \frac \cdot \frac + E_(x) + R_(x, T_v) + I_(x),$

where for any increasingly small $0 < \varepsilon < \frac-\alpha$ we define

:$I_(x) := \frac \int_^ \frac \cdot \frac ds,$

and where the remainder term

:$R_(x, T) \ll x^ + \frac + \frac,$

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 $L_(x)$ 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

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{{DEFAULTSORT:Liouville Function
Multiplicative functions