The function

(1) 
where is the number of not necessarily distinct Prime Factors of , with . The first few values of
are 1, , , 1, , 1, , , 1, 1, , , .... The Liouville function is connected with the
Riemann Zeta Function by the equation

(2) 
(Lehman 1960).
The Conjecture that the Summatory Function

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

(4) 
where
is the Floor Function (Lehman 1960). Lehman (1960) also gives the formulas



(5) 
and

(6) 
where , , and are variables ranging over the Positive integers, is the Möbius
Function, is Mertens Function, and , , and are Positive real numbers with
.
See also Pólya Conjecture, Prime Factors, Riemann Zeta Function
References
Fawaz, A. Y. ``The Explicit Formula for .'' Proc. London Math. Soc. 1, 86103, 1951.
Lehman, R. S. ``On Liouville's Function.'' Math. Comput. 14, 311320, 1960.
Sloane, N. J. A. Sequences
A028488 and
A002819/M0042
in ``An OnLine 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, 187189, 1980.
© 19969 Eric W. Weisstein
19990525