Editing Liouville function (section)

===Generalizations===

More generally, we can consider the weighted summatory functions over the Liouville function defined for any <math>\alpha \in \mathbb{R}</math> as follows for positive integers ''x'' where (as above) we have the special cases <math>L(x) := L_0(x)</math> and <math>T(x) = L_1(x)</math> <ref name="HUMPHRIES-WEIGHTED-SUMFUNCS"/>

:<math>L_{\alpha}(x) := \sum_{n \leq x} \frac{\lambda(n)}{n^{\alpha}}.</math> 

These <math>\alpha^{-1}</math>-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 <math>L(x)</math> precisely corresponds to the sum 

:<math>L(x) = \sum_{d^2 \leq x} M\left(\frac{x}{d^2}\right) = \sum_{d^2 \leq x} \sum_{n \leq \frac{x}{d^2}} \mu(n).</math> 

Moreover, these functions satisfy similar bounding asymptotic relations.<ref name="HUMPHRIES-WEIGHTED-SUMFUNCS"/> For example, whenever <math>0 \leq \alpha \leq \frac{1}{2}</math>, we see that there exists an absolute constant <math>C_{\alpha} > 0</math> such that 

:<math>L_{\alpha}(x) = O\left(x^{1-\alpha}\exp\left(-C_{\alpha} \frac{(\log x)^{3/5}}{(\log\log x)^{1/5}}\right)\right).</math>

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

:<math>\frac{\zeta(2\alpha+2s)}{\zeta(\alpha+s)} = s \cdot \int_1^{\infty} \frac{L_{\alpha}(x)}{x^{s+1}} dx,</math> 

which then can be inverted via the [[Mellin transform|inverse transform]] to show that for <math>x > 1</math>, <math>T \geq 1</math> and <math>0 \leq \alpha < \frac{1}{2}</math> 

:<math>L_{\alpha}(x) = \frac{1}{2\pi\imath} \int_{\sigma_0-\imath T}^{\sigma_0+\imath T} \frac{\zeta(2\alpha+2s)}{\zeta(\alpha+s)} 
     \cdot \frac{x^s}{s} ds + E_{\alpha}(x) + R_{\alpha}(x, T), </math>

where we can take <math>\sigma_0 := 1-\alpha+1 / \log(x)</math>, and with the remainder terms defined such that <math>E_{\alpha}(x) = O(x^{-\alpha})</math> and <math>R_{\alpha}(x, T) \rightarrow 0</math> as <math>T \rightarrow \infty</math>. 

In particular, if we assume that the 
[[Riemann hypothesis]] (RH) is true and that all of the non-trivial zeros, denoted by <math>\rho = \frac{1}{2} + \imath\gamma</math>, of the [[Riemann zeta function]] are [[simple zero|simple]], then for any <math>0 \leq \alpha < \frac{1}{2}</math> and <math> x \geq 1</math> there exists an infinite sequence of <math>\{T_v\}_{v \geq 1}</math> which satisfies that <math>v \leq T_v \leq v+1</math> for all ''v'' such that 

:<math>L_{\alpha}(x) = \frac{x^{1/2-\alpha}}{(1-2\alpha) \zeta(1/2)} + \sum_{|\gamma| < T_v} \frac{\zeta(2\rho)}{\zeta^{\prime}(\rho)} \cdot 
     \frac{x^{\rho-\alpha}}{(\rho-\alpha)} + E_{\alpha}(x) + R_{\alpha}(x, T_v) + I_{\alpha}(x), </math> 

where for any increasingly small <math>0 < \varepsilon < \frac{1}{2}-\alpha</math> we define 

:<math>I_{\alpha}(x) := \frac{1}{2\pi\imath \cdot x^{\alpha}} \int_{\varepsilon+\alpha-\imath\infty}^{\varepsilon+\alpha+\imath\infty} 
     \frac{\zeta(2s)}{\zeta(s)} \cdot \frac{x^s}{(s-\alpha)} ds,</math> 

and where the remainder term 

:<math>R_{\alpha}(x, T) \ll x^{-\alpha} + \frac{x^{1-\alpha} \log(x)}{T} + \frac{x^{1-\alpha}}{T^{1-\varepsilon} \log(x)}, </math> 

which of course tends to ''0'' as <math>T \rightarrow \infty</math>. These exact analytic formula expansions again share similar properties to those corresponding to the weighted [[Mertens function]] cases. Additionally, since <math>\zeta(1/2) < 0</math> we have another similarity in the form of <math>L_{\alpha}(x)</math> to <math>M(x)</math> 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''.