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

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

Template: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:

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

Möbius inversion of this formula yields

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

The Dirichlet inverse of 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

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

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

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

$L(n)=\sum _{k=1}^{n}\lambda (k)$ Template:OEIS,

the problem asks whether L(n) ≤ 0 for all 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, while it can also be shown via the same methods that L(n) < −1.3892783√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.

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 $T(x)=L_{1}(x)$

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

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

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

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 $M(x)$ 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.

References

• Pólya, G. (1919). "Verschiedene Bemerkungen zur Zahlentheorie". Jahresbericht der Deutschen Mathematiker-Vereinigung. 28: 31–40.
• Haselgrove, C. Brian (1958). "A disproof of a conjecture of Pólya". Mathematika. 5 (2): 141–145. doi:10.1112/S0025579300001480. ISSN 0025-5793. MR 0104638. Zbl 0085.27102.
• Lehman, R. (1960). "On Liouville's function". Mathematics of Computation. 14 (72): 311–320. doi:10.1090/S0025-5718-1960-0120198-5. MR 0120198.
• Tanaka, Minoru (1980). "A Numerical Investigation on Cumulative Sum of the Liouville Function". Tokyo Journal of Mathematics. 3 (1): 187–189. doi:10.3836/tjm/1270216093. MR 0584557.
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