Liouville function explained

The Liouville lambda function, denoted by and named after Joseph Liouville, is an important arithmetic function. Its value is if is the product of an even number of prime numbers, and if it is the product of an odd number of primes.

Explicitly, the fundamental theorem of arithmetic states that any positive integer can be represented uniquely as a product of powers of primes:, where are primes and the are positive integers. (is given by the empty product.) The prime omega functions count the number of primes, with or without multiplicity:

\omega(n) = k, \Omega(n) = a _{1} + a _{2} + … + a _{k.}

is defined by the formula

λ (n) = ( -1 ) ^{\Omega(n)}

.

is completely multiplicative since is completely additive, i.e.: . Since has no prime factors,, so .

It is related to the Möbius function . Write as, where is squarefree, i.e., . Then

λ (n) = \mu(b).

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

\sum _{d|n} λ (d) = \begin{cases} 1 & if n isaperfectsquare, \\ 0 & otherwise. \end{cases}

Möbius inversion of this formula yields

λ (n) = \sum _{d ^{2|n}} \mu\left( (n)/(d ^{2}) \right).

The Dirichlet inverse of Liouville function is the absolute value of the Möbius function,, the characteristic function of the squarefree integers. We also have that .

Series

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

(\zeta( 2 s))/(\zeta(s)) = \sum ^{inf ty} _{n =1} (λ (n))/(n ^{s}) .

Also:

\sum\limits ^{inf ty} _{n =1} (λ (n) ln n)/(=- \zeta( 2 ) =- n) (\pi ^{2})/(6) .

The Lambert series for the Liouville function is

\sum ^{inf ty} _{n =1} (λ (n)q ^{n})/(1- q ^{n}) = \sum ^{inf ty} _{n =1} q ^{n ^{2}} = (1)/(2) \left(\vartheta _{3(q)-1\right),}

where

\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) = \sum ^{n} _{k =1} λ (k) ,

the problem asks whether

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 for infinitely many positive integers n,^[1] 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{x} \exp\left(C ⋅ log ^{1/2} (x) \left( loglog x\right) ^{5/2+\varepsilon} \right)\right),

where the

C > 0 is some absolute limiting constant.^[2]

Define the related sum

T(n) = \sum ^{n} _{k =1} (λ (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, 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 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} (λ (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( (x)/(d ^{2}) \right) = \sum _{d ^{2} \leq x} \sum _{n \leq (x)/(d ^{2})} \mu(n).

Moreover, these functions satisfy similar bounding asymptotic relations.^[2] For example, whenever

0 \leq \alpha \leq (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} (( log x) ^{3/5})/(( loglog x) ^{1/5}) \right)\right).

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

(\zeta( 2 \alpha +2 s))/(\zeta(\alpha + s)) = s ⋅ \int ^{inf ty} _{1} (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 < (1)/(2) L _{\alpha} (x) = (1)/(2 \pi\imath) \int ^{\sigma _{0+\imath} T} _{\sigma _{0-\imath} T} (\zeta( 2 \alpha +2 s))/(\zeta(\alpha + s)) ⋅ (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) → 0 as T → inf ty .

In particular, if we assume that the Riemann hypothesis (RH) is true and that all of the non-trivial zeros, denoted by

\rho = (1)/(2) + \imath\gamma , of the Riemann zeta function are simple, then for any 0 \leq \alpha < (1)/(2) and x \geq 1 there exists an infinite sequence of \{T _{v\}} _{v} which satisfies that v \leq T _{v} \leq v +1 for all v such that L _{\alpha} (x) = (x ^{1/2-\alpha})/(( 1-2 \alpha) \zeta( 1 / 2 )) + \sum _{|\gamma| < T _{v}} (\zeta( 2 \rho))/(\zeta ^{\prime} (\rho)) ⋅ (x ^{\rho-\alpha})/((\rho - \alpha)) + E _{\alpha} (x) + R _{\alpha} (x, T _{v)} + I _{\alpha} (x),

where for any increasingly small

0 < \varepsilon < (1)/(2) - \alpha we define I _{\alpha} (x) : = (1)/(2 \pi\imath ⋅ x ^{\alpha}) \int ^{\varepsilon + \alpha + \imath inf ty} _{\varepsilon + \alpha - \imath inf ty} (\zeta( 2 s))/(\zeta(s)) ⋅ (x ^{s})/((s - \alpha)) ds,

and where the remainder term

R _{\alpha} (x, T) \ll x ^{-\alpha} + (x ^{1-\alpha} log (x))/(T) + (x ^{1-\alpha})/(T ^{1-\varepsilon} log (x)) ,

which of course tends to 0 as

T → inf ty . 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

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

Notes and References

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