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: $n=p_{1}^{a_{1}}\cdots p_{k}^{a_{k}}$ where p₁ < p₂ < ... < 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) = a₁ + a₂ + ... + 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 = a²b 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, $\lambda ^{-1}(n)=|\mu (n)|=\mu ^{2}(n),$ 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 {\zeta (2s)}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\lambda (n)}{n^{s}}}.

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

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.

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

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.

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

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

L(n)=\sum _{k=1}^{n}\lambda (k)

(sequence A002819 in the OEIS),

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√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 Lioville 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.

References

Borwein, P.; Ferguson, R.; Mossinghoff, M. J. (2008). "Sign Changes in Sums of the Liouville Function". Mathematics of Computation. 77 (263): 1681–1694. doi:10.1090/S0025-5718-08-02036-X.
Humphries, Peter (2013). "The distribution of weighted sums of the Liouville function and Pólyaʼs conjecture". Journal of Number Theory. 133 (2): 545–582. arXiv:1108.1524. doi:10.1016/j.jnt.2012.08.011.

Polya, G. (1919). "Verschiedene Bemerkungen zur Zahlentheorie". Jahresbericht der Deutschen Mathematiker-Vereinigung. 28: 31–40.
Haselgrove, C. Brian (1958). "A disproof of a conjecture of Polya". Mathematika. 5 (2): 141–145. doi:10.1112/S0025579300001480. ISSN 0025-5793. MR 0104638. Zbl 0085.27102.
Lehman, R. (1960). "On Liouville's function". Math. Comp. 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.
Weisstein, Eric W. "Liouville Function". MathWorld.
A.F. Lavrik (2001) [1994], "Liouville function", Encyclopedia of Mathematics, EMS Press

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[1] Borwein, P.; Ferguson, R.; Mossinghoff, M. J. (2008). "Sign Changes in Sums of the Liouville Function". Mathematics of Computation. 77 (263): 1681–1694. doi:10.1090/S0025-5718-08-02036-X.

[HUMPHRIES-WEIGHTED-SUMFUNCS-2] Humphries, Peter (2013). "The distribution of weighted sums of the Liouville function and Pólyaʼs conjecture". Journal of Number Theory. 133 (2): 545–582. arXiv:1108.1524. doi:10.1016/j.jnt.2012.08.011.