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Mertens Function
The Mertens function is the summatory function
 |
(1)
|
for nonnegative real numbers 
, where the sum is over positive
integers and 
is the Möbius function (Mertens 1897; Havil
2003, p. 208). For 
, 2, ..., 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, ... |
Titchmarsh (1951) 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)
|
for 
,
where 
is the floor function (Lehman 1960).
Mertens (1897) verified that 
for 
and conjectured that
 |
(6)
|
for all 
.
This Mertens conjecture was disproved by Odlyzko
and te Riele (1985).
Lehman (1960) gives an algorithm for computing 
with 
operations for every 
, while the Lagarias-Odlyzko (1987) algorithm
for computing the prime counting function 
can be modified to give 
in 
operations for every 
. Deléglise and Rivat (1996) described an
elementary method for computing isolated values of 
with time complexity 
and space complexity 
.
See also
Mertens Conjecture, Möbius Function, Redheffer Matrix, SquarefreeExplore with Wolfram|Alpha
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. "Ueber
einige asymptotische Gesetze der Zahlentheorie." J. reine angew. Math. 77,
289-338, 1874. https://doi.org/10.1515/crll.1874.77.289.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 the Riemann Zeta-Function. Oxford, England: Clarendon
Press, 1951.Referenced on Wolfram|Alpha
Mertens FunctionCite this as:
Weisstein, Eric W. "Mertens Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MertensFunction.html