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.
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,
![M_0(x)=<span style=]() {M(x)-1/2mu(x) if x in Z^+; M(x) otherwise, " src="https://mathworld.wolfram.com/images/equations/MertensFunction/NumberedEquation3.gif" style="height:48px; width:222px" /> |
(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 
.
REFERENCES:
Deléglise, M. and Rivat, J. "Computing the Summation of the Möbius Function." Experiment. Math. 5, 291-295, 1996.
Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, p. 250, 2004.
Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 208-210, 2003.
Lagarias, J. and Odlyzko, A. "Computing 
: An Analytic Method." J. Algorithms 8, 173-191, 1987.
Lehman, R. S. "On Liouville's Function." Math. Comput. 14, 311-320, 1960.
Lehmer, D. H. Guide to Tables in the Theory of Numbers. Bulletin No. 105. Washington, DC: National Research Council, pp. 7-10, 1941.
Mertens, F. "Über einige asymptotische Gesetze der Zahlentheorie." J. reine angew. Math. 77, 46-62, 1874.
Mertens, F. "Über eine zahlentheoretische Funktion." Akad. Wiss. Wien Math.-Natur. Kl. Sitzungsber. IIa 106, 761-830, 1897.
Odlyzko, A. M. and te Riele, H. J. J. "Disproof of the Mertens Conjecture." J. reine angew. Math. 357, 138-160, 1985.
Sloane, N. J. A. Sequences A002321/M0102, A028442, A084237, and A118684 in "The On-Line Encyclopedia of Integer Sequences."
Sterneck, R. D. von. "Empirische Untersuchung über den Verlauf der zahlentheoretischer Function 
im Intervalle von 0 bis 150 000." Sitzungsber. der Kaiserlichen Akademie der Wissenschaften Wien, Math.-Naturwiss. Klasse 2a 106, 835-1024, 1897.
Titchmarsh, E. C. The Theory of Functions, 2nd ed. Oxford, England: Oxford University Press, 1960.
