The Liouville summatory function is the summatory function
 |
(1)
|
where 
is the Liouville function. Its first values
for 
,
2, ... are 1, 0, 
, 0, 
, 0, 
, 
, 
, 0, 
, 
, 
, 
, 
, 0, 
, 
, 
, 
, ... (OEIS A002819).
Lehman (1960) gives the formulas
 |
(2)
|
and
 |
(3)
|
where 
,
,
and 
range over the positive integers, 
is the Möbius function,
is the Mertens function, and 
, 
, and 
are positive real
numbers with 
.
The refuted Pólya conjecture asserted that 
for all 
.
The first positive value occurs at 
(Tanaka 1980). The first zeros are 2, 4, 6, 10,
16, 26, 40, 96, 586, 906150256, 906150294, 906150308, 906150310, 906150314, 906151516,
... (OEIS A028488). It is not known whether
changes sign infinitely often.
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, 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.