The function
 |
(1)
|
where 
is the number of not necessarily distinct prime factors
of 
,
with 
.
The values of 
for 
,
2, ... are 1, 
,
,
1, 
,
1, 
,
,
1, 1, 
,
,
... (OEIS A008836). The values of 
such that 
are 2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20,
23, ... (OEIS A026424), while then values such
that 
are 1, 4, 6, 9, 10, 14, 15, 16, 21, 22, 24, ... (OEIS A028260).
The Liouville function is implemented in the Wolfram Language as LiouvilleLambda[n].
The Liouville function is connected with the Riemann zeta function by the equation
 |
(2)
|
(Lehman 1960). It has the Lambert series
 |  |  |
(3)
|
 |  | ![1/2[theta_3(q)-1],](/images/equations/LiouvilleFunction/Inline21.svg) |
(4)
|
where 
is a Jacobi theta function.
Consider the summatory function
 |
(5)
|
the values of which for 
, 2, ... are 1, 0, 
, 0, 
, 0, 
, 
, 
, 0, 
, 
, 
, 
, 
, 0, 
, 
, 
, 
, ... (OEIS A002819).
Lehman (1960) gives the formulas
 |
(6)
|
and
 |
(7)
|
where 
,
,
and 
are variables ranging over the positive integers,
is the Möbius function, 
is Mertens function,
and 
,
,
and 
are positive real numbers with 
.
The conjecture that 
satisfies 
for 
is called the Pólya
conjecture and has been proved to be false. 
is positive for 
, but not for any other 
for a long time. In fact, the first 
for which 
are for 
, 4, 6, 10, 16, 26, 40, 96, 586, 906150256, ... (OEIS A028488), and 
is the first counterexample to the Pólya
conjecture (Tanaka 1980). However, it is unknown if 
changes sign infinitely often (Tanaka 1980).
The values of 
for 
,
1, 2, ... are 1, 0, 
,
,
,
,
,
,
,
... (OEIS A090410).
." Proc. London Math. Soc. 1, 86-103,
1951.Gupta, H. "On a Table of Values of 
." Proc. Indian Acad. Sci. Sec. A 12,
407-409, 1940.Gupta, H. "A Table of Values of Liouville's Function
."
Res. Bull. East Panjab University, No. 3, 45-55, Feb. 1950.Lehman,
R. S. "On Liouville's Function." Math. Comput. 14, 311-320,
1960.Ramanujan, S. "Irregular Numbers." J. Indian Math.
Soc. 5, 105-106, 1913. Ramanujan, S.