The Liouville summatory function is the summatory function
 |
(1)
|
where 
is a positive integer and 
is the Liouville
function. For real numbers 
, define 
, with 
, where 
is the floor function.
Its first values for 
, 2, ... are 1, 0, 
, 0, 
, 0, 
, 
, 
, 0, 
, 
, 
, 
, 
, 0, 
, 
, 
, 
, ... (OEIS A002819).
Lehman (1960) gives the following formulas (see also Borwein et al. 2008).
 |
(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 second formula holds for any positive
integer 
.
The refuted Pólya conjecture asserted that 
for all 
.
In this range, 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).
The function 
changes sign infinitely often (Borwein et al. 2008).
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.