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Liouville function
Arithmetic 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 = p1a1 ⋯ pkak, where p1 < p2 < ... < pk are primes and the aj 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 , {\displaystyle \omega (n)=k,} Ω ( n ) = a 1 + a 2 + ⋯ + a k . {\displaystyle \Omega (n)=a_{1}+a_{2}+\cdots +a_{k}.}

λ(n) is defined by the formula

λ ( n ) = ( − 1 ) Ω ( n ) {\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

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

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

∑ d | n λ ( d ) = { 1 if  n  is a perfect square, 0 otherwise. {\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

λ ( n ) = ∑ d 2 | n μ ( n d 2 ) . {\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, λ–1(n) = |μ(n)| = μ2(n), the characteristic function of the squarefree integers.

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Series

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

ζ ( 2 s ) ζ ( s ) = ∑ n = 1 ∞ λ ( n ) n s . {\displaystyle {\frac {\zeta (2s)}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\lambda (n)}{n^{s}}}.}

Also:

∑ n = 1 ∞ λ ( n ) ln ⁡ n n = − ζ ( 2 ) = − π 2 6 . {\displaystyle \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

∑ n = 1 ∞ λ ( n ) q n 1 − q n = ∑ n = 1 ∞ q n 2 = 1 2 ( ϑ 3 ( q ) − 1 ) , {\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 ϑ 3 ( q ) {\displaystyle \vartheta _{3}(q)} is the Jacobi theta function.

Conjectures on weighted summatory functions

The Pólya problem is a question raised made by George Pólya in 1919. Defining

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

the problem asks whether L ( n ) ≤ 0 {\displaystyle 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 ε > 0 {\displaystyle \varepsilon >0} , assuming the Riemann hypothesis, we have that the summatory function L ( x ) ≡ L 0 ( x ) {\displaystyle L(x)\equiv L_{0}(x)} is bounded by

L ( x ) = O ( x exp ⁡ ( C ⋅ log 1 / 2 ⁡ ( x ) ( log ⁡ log ⁡ x ) 5 / 2 + ε ) ) , {\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 C > 0 {\displaystyle C>0} is some absolute limiting constant.3

Define the related sum

T ( n ) = ∑ k = 1 n λ ( k ) k . {\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 Liouville function defined for any α ∈ R {\displaystyle \alpha \in \mathbb {R} } as follows for positive integers x where (as above) we have the special cases L ( x ) := L 0 ( x ) {\displaystyle L(x):=L_{0}(x)} and T ( x ) = L 1 ( x ) {\displaystyle T(x)=L_{1}(x)} 4

L α ( x ) := ∑ n ≤ x λ ( n ) n α . {\displaystyle L_{\alpha }(x):=\sum _{n\leq x}{\frac {\lambda (n)}{n^{\alpha }}}.}

These α − 1 {\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-weighted, or ordinary function L ( x ) {\displaystyle L(x)} precisely corresponds to the sum

L ( x ) = ∑ d 2 ≤ x M ( x d 2 ) = ∑ d 2 ≤ x ∑ n ≤ x d 2 μ ( n ) . {\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, these functions satisfy similar bounding asymptotic relations.5 For example, whenever 0 ≤ α ≤ 1 2 {\displaystyle 0\leq \alpha \leq {\frac {1}{2}}} , we see that there exists an absolute constant C α > 0 {\displaystyle C_{\alpha }>0} such that

L α ( x ) = O ( x 1 − α exp ⁡ ( − C α ( log ⁡ x ) 3 / 5 ( log ⁡ log ⁡ x ) 1 / 5 ) ) . {\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

ζ ( 2 α + 2 s ) ζ ( α + s ) = s ⋅ ∫ 1 ∞ L α ( x ) x s + 1 d x , {\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 x > 1 {\displaystyle x>1} , T ≥ 1 {\displaystyle T\geq 1} and 0 ≤ α < 1 2 {\displaystyle 0\leq \alpha <{\frac {1}{2}}}

L α ( x ) = 1 2 π ı ∫ σ 0 − ı T σ 0 + ı T ζ ( 2 α + 2 s ) ζ ( α + s ) ⋅ x s s d s + E α ( x ) + R α ( x , T ) , {\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 σ 0 := 1 − α + 1 / log ⁡ ( x ) {\displaystyle \sigma _{0}:=1-\alpha +1/\log(x)} , and with the remainder terms defined such that E α ( x ) = O ( x − α ) {\displaystyle E_{\alpha }(x)=O(x^{-\alpha })} and R α ( x , T ) → 0 {\displaystyle R_{\alpha }(x,T)\rightarrow 0} as T → ∞ {\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 ρ = 1 2 + ı γ {\displaystyle \rho ={\frac {1}{2}}+\imath \gamma } , of the Riemann zeta function are simple, then for any 0 ≤ α < 1 2 {\displaystyle 0\leq \alpha <{\frac {1}{2}}} and x ≥ 1 {\displaystyle x\geq 1} there exists an infinite sequence of { T v } v ≥ 1 {\displaystyle \{T_{v}\}_{v\geq 1}} which satisfies that v ≤ T v ≤ v + 1 {\displaystyle v\leq T_{v}\leq v+1} for all v such that

L α ( x ) = x 1 / 2 − α ( 1 − 2 α ) ζ ( 1 / 2 ) + ∑ | γ | < T v ζ ( 2 ρ ) ζ ′ ( ρ ) ⋅ x ρ − α ( ρ − α ) + E α ( x ) + R α ( x , T v ) + I α ( x ) , {\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 0 < ε < 1 2 − α {\displaystyle 0<\varepsilon <{\frac {1}{2}}-\alpha } we define

I α ( x ) := 1 2 π ı ⋅ x α ∫ ε + α − ı ∞ ε + α + ı ∞ ζ ( 2 s ) ζ ( s ) ⋅ x s ( s − α ) d s , {\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

R α ( x , T ) ≪ x − α + x 1 − α log ⁡ ( x ) T + x 1 − α T 1 − ε log ⁡ ( x ) , {\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 T → ∞ {\displaystyle T\rightarrow \infty } . These exact analytic formula expansions again share similar properties to those corresponding to the weighted Mertens function cases. Additionally, since ζ ( 1 / 2 ) < 0 {\displaystyle \zeta (1/2)<0} we have another similarity in the form of L α ( x ) {\displaystyle L_{\alpha }(x)} to M ( x ) {\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. 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. https://doi.org/10.1090%2FS0025-5718-08-02036-X ↩

  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. https://doi.org/10.1016%2Fj.jnt.2012.08.011 ↩

  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. https://doi.org/10.1016%2Fj.jnt.2012.08.011 ↩

  4. 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. https://doi.org/10.1016%2Fj.jnt.2012.08.011 ↩

  5. 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. https://doi.org/10.1016%2Fj.jnt.2012.08.011 ↩