FUNCIÓN : DE LIOUVILLE .- ......................:http://es.wikipedia.org/w/index.php?title=Funci%C3%B3n_de_Liouville&printable=yes
FUNCIÓN DE MERTENS .- .....................:http://es.wikipedia.org/w/index.php?title=Funci%C3%B3n_de_Mertens&printable=yes
Mertens Function
The Mertens function is the summary function
 |
(1)
|
where 
is the Möbius function (Mertens 1897; Havil 2003, p. 208). The first few values are 1, 0, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, ... (OEIS A002321). 
is also given by the determinant of the 
Redheffer matrix.
is the Möbius function (Mertens 1897; Havil 2003, p. 208). The first few values are 1, 0, , , , , , , , , , , ... (OEIS A002321). is also given by the determinant of the Redheffer matrix.
Values of 
for 
, 1, 2, ... are given by 1, 
, 1, 2, 
, 
, 212, 1037, 1928, 
, ... (OEIS A084237; Deléglise and Rivat 1996).
for , 1, 2, ... are given by 1, , 1, 2, , , 212, 1037, 1928, , ... (OEIS A084237; Deléglise and Rivat 1996).
The following table summarizes the first few values of 
at which 
for various 
at which for various  | Sloane |  such that  |
 | 13, 19, 20, 30, 33, 43, 44, 45, 47, 48, 49, 50, ... | |
 | 5, 7, 8, 9, 11, 12, 14, 17, 18, 21, 23, 24, 25, 29, ... | |
 | 3, 4, 6, 10, 15, 16, 22, 26, 27, 28, 35, 36, 38, ... | |
| 0 | A028442 | 2, 39, 40, 58, 65, 93, 101, 145, 149, 150, ... |
| 1 | A118684 | 1, 94, 97, 98, 99, 100, 146, 147, 148, 161, ... |
| 2 | 95, 96, 217, 229, 335, 336, 339, 340, 345, 347, 348, ... | |
| 3 | 218, 223, 224, 225, 227, 228, 341, 342, 343, 344, 346, ... |
An analytic formula for 
is not known, although Titchmarsh (1960) showed that if the Riemann hypothesisholds and if there are no multiple Riemann zeta function zeros, then there is a sequence 
with 
such that
is not known, although Titchmarsh (1960) showed that if the Riemann hypothesisholds and if there are no multiple Riemann zeta function zeros, then there is a sequence with such that |
(2)
|
 |
(3)
|
and 
runs over all nontrivial zeros of the Riemann zeta function (Odlyzko and te Riele 1985).
runs over all nontrivial zeros of the Riemann zeta function (Odlyzko and te Riele 1985).
The Mertens function is related to the number of squarefree integers up to 
, which is the sum from 1 to 
of the absolute value of 
,
, which is the sum from 1 to of the absolute value of , |
(4)
|
The Mertens function also obeys
 |
(5)
|
(Lehman 1960).
Mertens (1897) verified that 
for 
and conjectured that this inequality holds for all nonnegative 
. The statement
for and conjectured that this inequality holds for all nonnegative . The statement |
(6)
|
is therefore known as the Mertens conjecture, although it has since been disproved.
Lehman (1960) gives an algorithm for computing 
with 
operations, while the Lagarias-Odlyzko (1987) algorithm for computing the prime counting function 
can be modified to give 
in 
operations. Deléglise and Rivat 1996) described an elementary method for computing isolated values of 
with time complexity 
and space complexity 
.
with operations, while the Lagarias-Odlyzko (1987) algorithm for computing the prime counting function can be modified to give in operations. Deléglise and Rivat 1996) described an elementary method for computing isolated values of with time complexity and space complexity .
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