Generalizations
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 and [2]
These -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 precisely corresponds to the sum
Moreover, these functions satisfy similar bounding asymptotic relations.[2] 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 (inverse) 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 (RH) 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 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.