Liouville function

The Liouville lambda function, denoted by λ(n) and named after Joseph Liouville, is an important arithmetic function.

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.

The prime omega functions count the number of primes, with (Ω) or without (ω) multiplicity: λ(n) is defined by the formula (sequence A008836 in the OEIS).

Since 1 has no prime factors, Ω(1) = 0, so λ(1) = 1.

It is related to the Möbius function μ(n).

Then The sum of the Liouville function over the divisors of n is the characteristic function of the squares: Möbius inversion of this formula yields 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.

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.

Defining the problem asks whether

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

, assuming the Riemann hypothesis, we have that the summatory function

is some absolute limiting constant.

[2] 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 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.

More generally, we can consider the weighted summatory functions over the Liouville function defined for any

as follows for positive integers x where (as above) we have the special cases

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.

, we see that there exists an absolute constant

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 to show that for

, and with the remainder terms defined such that

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

, of the Riemann zeta function are simple, then for any

there exists an infinite sequence of

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.

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.