Liouville Function

The function

$$\lambda(n)=(-1)^{r(n)},$$ (1)

where $r(n)$ is the number of not necessarily distinct Prime Factors of $n$, with $r(1)=0$. The first few values of $\lambda(n)$ are 1, $-1$, $-1$, 1, $-1$, 1, $-1$, $-1$, 1, 1, $-1$, $-1$, .... The Liouville function is connected with the Riemann Zeta Function by the equation

$${\zeta(2s)\over\zeta(s)}=\sum_{n=1}^\infty {\lambda(n)\over n^s}$$ (2)

(Lehman 1960).

The Conjecture that the Summatory Function

$$L(n)\equiv \sum_{k=1}^n \lambda(n)$$ (3)

satisfies $L(n)\leq 0$ for $n\geq 2$ is called the Pólya Conjecture and has been proved to be false. The first $n$ for which $L(n)=0$ are for $n=2$, 4, 6, 10, 16, 26, 40, 96, 586, 906150256, ... (Sloane's A028488), and $n=906150257$ is, in fact, the first counterexample to the Pólya Conjecture (Tanaka 1980). However, it is unknown if $L(x)$ changes sign infinitely often (Tanaka 1980). The first few values of $L(n)$ are 1, 0, $-1$, 0, $-1$, 0, $-1$, $-2$, $-1$, 0, $-1$, $-2$, $-3$, $-2$, $-1$, 0, $-1$, $-2$, $-3$, $-4$, ... (Sloane's A002819). $L(n)$ also satisfies

$$\sum_{n=1}^x L\left({x\over n}\right)=\left\lfloor{\sqrt{x}}\right\rfloor ,$$ (4)

where $\left\lfloor{x}\right\rfloor$ is the Floor Function (Lehman 1960). Lehman (1960) also gives the formulas

$L(x)=\sum_{m=1}^{x/w} \mu(m)\left\{{\left\lfloor{\sqrt{x\over m}}\right\rfloor ... ...x\over km}\right\rfloor -\left\lfloor{x\over mv}\right\rfloor }\right)}\right\}$
$-\sum_{l=x/w-1}^{x/v} L\left({x\over l}\right)\sum_{\scriptstyle m\vert l\atop\scriptstyle m=1}^{x/w} \mu(m)\quad$	(5)

and

$$L(x)=\sum_{k=1}^g M\left({x\over k^2}\right)+\sum_{l=1}^{x/g... ...over g^2}\right)\left\lfloor{\sqrt{x\over g^2}}\right\rfloor ,$$ (6)

where $k$, $l$, and $m$ are variables ranging over the Positive integers, $\mu(n)$ is the Möbius Function, $M(x)$ is Mertens Function, and $v$, $w$, and $x$ are Positive real numbers with $v<w<x$.

See also Pólya Conjecture, Prime Factors, Riemann Zeta Function

References

Fawaz, A. Y. ``The Explicit Formula for $L_0(x)$.'' Proc. London Math. Soc. 1, 86-103, 1951.

Lehman, R. S. ``On Liouville's Function.'' Math. Comput. 14, 311-320, 1960.

Sloane, N. J. A. Sequences A028488 and A002819/M0042 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Tanaka, M. ``A Numerical Investigation on Cumulative Sum of the Liouville Function.'' Tokyo J. Math. 3, 187-189, 1980.