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:   ${\displaystyle n=p_{1}^{a_{1}}\cdots p_{k}^{a_{k}}}$   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

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

${\displaystyle \lambda (n)=\mu (b).}$

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

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

Möbius inversion of this formula yields

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

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

Series

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

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

The Lambert series for the Liouville function is

${\displaystyle \sum _{n=1}^{\infty }{\frac {\lambda (n)q^{n}}{1-q^{n}}}=\sum _{n=1}^{\infty }q^{n^{2}}={\frac {1}{2}}\left(\vartheta _{3}(q)-1\right),}$

where ${\displaystyle \vartheta _{3}(q)}$ 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

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

the conjecture states that ${\displaystyle L(n)\leq 0}$ 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 ${\displaystyle \varepsilon >0}$, assuming the Riemann hypothesis, we have that the summatory function ${\displaystyle L(x)\equiv L_{0}(x)}$ is bounded by

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

where the ${\displaystyle C>0}$ is some absolute limiting constant [3].

Define the related sum

${\displaystyle T(n)=\sum _{k=1}^{n}{\frac {\lambda (k)}{k}}.}$

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 ${\displaystyle \alpha \in \mathbb {R} }$ as follows for positive integers x where (as above) we have the special cases ${\displaystyle L(x):=L_{0}(x)}$ and ${\displaystyle T(x)=L_{1}(x)}$ [3]

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

These ${\displaystyle \alpha ^{-1}}$-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-weigthed, or ordinary function ${\displaystyle L(x)}$ precisely corresponds to the sum

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

Moreover, as noted in [3] these functions satisfy similar bounding asymptotic relations. For example, whenever ${\displaystyle 0\leq \alpha \leq {\frac {1}{2}}}$, we see that there exists an absolute constant ${\displaystyle C_{\alpha }>0}$ such that

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

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

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

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

${\displaystyle L_{\alpha }(x)={\frac {1}{2\pi \imath }}\int _{\sigma _{0}-\imath T}^{\sigma _{0}+\imath T}{\frac {\zeta (2\alpha +2s)}{\zeta (\alpha +s)}}\cdot {\frac {x^{s}}{s}}ds+E_{\alpha }(x)+R_{\alpha }(x,T),}$

where we can take ${\displaystyle \sigma _{0}:=1-\alpha +1/\log(x)}$, and with the remainder terms defined such that ${\displaystyle E_{\alpha }(x)=O(x^{-\alpha })}$ and ${\displaystyle R_{\alpha }(x,T)\rightarrow 0}$ as ${\displaystyle T\rightarrow \infty }$.

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

${\displaystyle L_{\alpha }(x)={\frac {x^{1/2-\alpha }}{(1-2\alpha )\zeta (1/2)}}+\sum _{|\gamma |<T_{v}}{\frac {\zeta (2\rho )}{\zeta ^{\prime }(\rho )}}\cdot {\frac {x^{\rho -\alpha }}{(\rho -\alpha )}}+E_{\alpha }(x)+R_{\alpha }(x,T_{v})+I_{\alpha }(x),}$

where for any increasingly small ${\displaystyle 0<\varepsilon <{\frac {1}{2}}-\alpha }$ we define

${\displaystyle I_{\alpha }(x):={\frac {1}{2\pi \imath \cdot x^{\alpha }}}\int _{\varepsilon +\alpha -\imath \infty }^{\varepsilon +\alpha +\imath \infty }{\frac {\zeta (2s)}{\zeta (s)}}\cdot {\frac {x^{s}}{(s-\alpha )}}ds,}$

and where the remainder term

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

which of course tends to 0 as ${\displaystyle T\rightarrow \infty }$. These exact analytic formula expansions again share similar properties to those corresponding to the weighted Mertens function cases. Additionally, since ${\displaystyle \zeta (1/2)<0}$ we have another similarity in the form of ${\displaystyle L_{\alpha }(x)}$ to ${\displaystyle M(x)}$ 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

1. P. Borwein, R. Ferguson, and M. J. Mossinghoff, Sign Changes in Sums of the Liouville Function, Mathematics of Computation 77 (2008), no. 263, 1681–1694.
2. Peter Humphries, The distribution of weighted sums of the Liouville function and Pólya’s conjecture, Journal of Number Theory 133 (2013), 545–582.
3. 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.CS1 maint: ref=harv (link)
• 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