In number theory, the Liouville function, named after French mathematician Joseph Liouville, is an important arithmetic function is the product of an even number of prime numbers 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:

1. ,

are positive integers. The prime omega function with multiplicity:

1. .

Thus, the Liouville function is defined by

.

Properties

is completely additive is completely multiplicative.

is also related to the Möbius function is squarefree, then

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

Möbius inversion of this formula yields

The Dirichlet inverse of the Liouville function is the absolute value, 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

is the Jacobi theta function.

Conjectures on weighted summatory functions

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

1. ,

the problem asks whether L(n)≤0 for all 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, while it can also be shown via the same methods that L(n)<−1.3892783 for infinitely many positive integers n.

is bounded by

is some absolute limiting constant.

Define the related sum

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

as follows for positive integers x

-weighted summatory functions are related to the Mertens function, or weighted summatory functions of the Möbius function precisely corresponds to the sum

Moreover, these functions satisfy similar bounding asymptotic relations. such that

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

which then can be inverted via the inverse transform

.

In particular, if we assume that the Riemann hypothesis, of the Riemann zeta function are simple for all v such that

we define

and where the remainder term

which of course tends to 0. These exact analytic formula expansions again share similar properties to those corresponding to the weighted Mertens function 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.

References