## Liouville Function

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, 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.

© 1996-9 Eric W. Weisstein
1999-05-25