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.
Values of 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
OEIS | 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 hypothesis holds and if there are no multiple Riemann zeta function zeros, then there is a sequence with such that
(2)
|
where is the Riemann zeta function,
(3)
|
and 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 ,
(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
(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 .