where
is the Möbius function, Stieltjes claimed
in an 1885 letter to Hermite that stays within two fixed bounds, which he suggested
could probably be taken to be (Havil 2003, p. 208). In the same year, Stieltjes
(1885) claimed that he had a proof of the general result. However, it seems likely
that Stieltjes was mistaken in this claim (Derbyshire 2004, pp. 160-161). Mertens
(1897) subsequently published a paper opining based on a calculation of that Stieltjes' claim
(2)
for
was "very probable."
The Mertens conjecture has important implications, since the truth of any equality
of the form
(3)
for any fixed
(the form of the Mertens conjecture with ) would imply the Riemann
hypothesis. In fact, the statement
(4)
for any
is equivalent to the Riemann hypothesis (Derbyshire
2004, p. 251).
Mertens (1897) verified the conjecture for , and this was subsequently extended to by von Sterneck (1912; Deléglise and Rivat
1996). The Mertens conjecture was proved false by Odlyzko and te Riele (1985). Their
proof is indirect and does not produce a specific counterexample, but it does show
that
(5)
(6)
(Havil 2003, p. 209). Odlyzko and te Riele (1985) believe that there are no counterexamples to the Mertens conjecture for , or even , calling Stieltjes' supposed proof into very strong
question (Derbyshire 2004, p. 161).
Pintz (1987) subsequently showed that at least one counterexample to the conjecture occurs for
(Havil 2003, p. 209), using a weighted integral average of and a discrete sum involving nontrivial zeros of the
Riemann zeta function. The first value of
for which
is still unknown, but it is known to exceed (te Riele 2006), improving the previous best results
of
(Lioen and van de Lune 1994) and (Dress 1993; Deléglise and Rivat 1996).
It is still not known if
(7)
although it seems very probable (Odlyzko and te Riele 1985).
Anderson, R. J. "On the Mertens Conjecture for Cusp Forms." Mathematika26, 236-249, 1979.Anderson,
R. J. "Corrigendum: 'On the Mertens Conjecture for Cusp Forms.' "
Mathematika27, 261, 1980.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, 2004.Devlin, K. "The Mertens Conjecture."
Irish Math. Soc. Bull.17, 29-43, 1986.Dress, F. "Fonction
sommatoire de la fonction de Möbius; 1. Majorations expérimentales."
Experiment. Math.2, 93-102, 1993.Grupp, F. "On the
Mertens Conjecture for Cusp Forms." Mathematika29, 213-226, 1982.Hardy,
G. H. Ramanujan:
Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York:
Chelsea, p. 64, 1999.Havil, J. Gamma:
Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Jurkat,
W. and Peyerimhoff, A. "A Constructive Approach to Kronecker Approximation and
Its Application to the Mertens Conjecture." J. reine angew. Math.286/287,
322-340, 1976.Lehman, R. S. "On Liouville's Functions."
Math. Comput.14, 311-320, 1960.Lioen, W. M. and
van de Lune, J. "Systematic Computations on Mertens' Conjecture and Dirichlet's
Divisor Problem by Vectorized Sieving." In From Universal Morphisms to Megabytes:
A Baayen Space Odyssey. On the Occasion of the Retirement of P. C. Baayen
(Ed. K. Apt, L. Schrijver, and N. Temme). Amsterdam, Netherlands:
Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, pp. 421-432,
1994. http://walter.lioen.com/papers/LL94.pdf.Mertens,
F. "Über eine zahlentheoretische Funktion." Sitzungsber. Akad.
Wiss. Wien IIa106, 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.Pintz, J. "An
Effective Disproof of the Mertens Conjecture." Astérique147-148,
325-333 and 346, 1987.Stieltjes, T. J. "Sur une loi asymptotique
dans la théorie des nombres." C. R. Acad. Sci. Paris101,
368-370, 1885.te Riele, H. J. J. "Some Historical and
Other Notes About the Mertens Conjecture and Its Recent Disproof." Nieuw
Arch. Wisk.3, 237-243, 1985.te Riele, H. R. "The
Mertens Conjecture Revisited." 7th Algorithmic Number Theory Symposium. Technische
Universität Berlin, 23-28 July 2006.http://www.math.tu-berlin.de/~kant/ants/Proceedings/te_riele/te_riele_talk.pdf.von
Sterneck, R. D. "Die zahlentheoretische Funktion bis zur Grenze 500000." Akad. Wiss. Wien Math.-Natur.
Kl. Sitzungsber. IIa121, 1083-1096, 1912.
is the Möbius function, Stieltjes claimed
in an 1885 letter to Hermite that 
stays within two fixed bounds, which he suggested
could probably be taken to be 
(Havil 2003, p. 208). In the same year, Stieltjes
(1885) claimed that he had a proof of the general result. However, it seems likely
that Stieltjes was mistaken in this claim (Derbyshire 2004, pp. 160-161). Mertens
(1897) subsequently published a paper opining based on a calculation of 
that Stieltjes' claim
was "very probable."
(the form of the Mertens conjecture with 
) would imply the Riemann
hypothesis. In fact, the statement
is equivalent to the Riemann hypothesis (Derbyshire
2004, p. 251).
, and this was subsequently extended to 
by von Sterneck (1912; Deléglise and Rivat
1996). The Mertens conjecture was proved false by Odlyzko and te Riele (1985). Their
proof is indirect and does not produce a specific counterexample, but it does show
that
, or even 
, calling Stieltjes' supposed proof into very strong
question (Derbyshire 2004, p. 161).
(Havil 2003, p. 209), using a weighted integral average of 
and a discrete sum involving nontrivial zeros of the
Riemann zeta function. The first value of
for which 
is still unknown, but it is known to exceed 
(te Riele 2006), improving the previous best results
of 
(Lioen and van de Lune 1994) and 
(Dress 1993; Deléglise and Rivat 1996).
bis zur Grenze 500000." Akad. Wiss. Wien Math.-Natur.
Kl. Sitzungsber. IIa 121, 1083-1096, 1912.