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).
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