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,

(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.