Liouville function


In number theory, the Liouville function, named after French mathematician Joseph Liouville and denoted, 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 prime numbers.

Definition

By the fundamental theorem of arithmetic, any positive integer can be represented uniquely as a product of powers of primes:
where are primes and the exponents are positive integers. The prime omega function counts the number of primes in the factorization of with multiplicity:
Thus, the Liouville function is defined by

Properties

Since is completely additive; i.e.,, then is completely multiplicative. Since has no prime factors,, so.
is also related to the Möbius function : if we write as, where is squarefree, then
The sum of the Liouville function over the divisors of is the characteristic function of the squares:
Möbius inversion of this formula yields
The Dirichlet inverse of the Liouville function is the absolute value of the Möbius function,, the characteristic function of the squarefree integers.

Series

The Dirichlet series for the Liouville function is related to the Riemann zeta function by
Also:
The Lambert series for the Liouville function is
where is the Jacobi theta function.

Conjectures on weighted summatory functions


[Image:Liouville.svg|thumb|none|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.]
Image:Liouville-big.svg|thumb|none|Summatory Liouville function L up to n = 107. Note the apparent scale invariance of the oscillations.
Image:Liouville-log.svg|thumb|none|Logarithmic graph of the negative of the summatory Liouville function L up to n = 2 × 109. The green spike shows the function itself in the narrow region where the Pólya conjecture fails; the blue curve shows the oscillatory contribution of the first Riemann zero.
[Image:Liouville-harmonic.svg|thumb|none|Harmonic Summatory Liouville function T(n) up to n = 103]

The Pólya problem is a question raised made by George Pólya in 1919. Defining
the problem asks whether for some n > 1. The answer turns out to be yes. The smallest counter-example is n = 906150257, found by Minoru Tanaka in 1980. It has since been shown that L > 0.0618672 for infinitely many positive integers n, while it can also be shown via the same methods that L < −1.3892783 for infinitely many positive integers n.
For any, assuming the Riemann hypothesis, we have that the summatory function is bounded by
where the is some absolute limiting constant.
Define the related sum
It was open for some time whether T ≥ 0 for sufficiently big n ≥ n0. This was then disproved by, who showed that T 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 as follows for positive integers x where we have the special cases and
These -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 precisely corresponds to the sum
Moreover, these functions satisfy similar bounding asymptotic relations. For example, whenever, we see that there exists an absolute constant such that
By an application of Perron's formula, or equivalently by a key Mellin transform, we have that
which then can be inverted via the inverse transform to show that for, and
where we can take, and with the remainder terms defined such that and as.
In particular, if we assume that the Riemann hypothesis is true and that all of the non-trivial zeros, denoted by, of the Riemann zeta function are simple, then for any and there exists an infinite sequence of which satisfies that for all v such that
where for any increasingly small we define
and where the remainder term
which of course tends to 0 as. These exact analytic formula expansions again share similar properties to those corresponding to the weighted Mertens function cases. Additionally, since we have another similarity in the form of to 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.