From: james@quanset.com (John Ames) Subject: Re: conjectures that are true for large n but are actually false Date: Sat, 23 Jan 1999 19:34:34 GMT Newsgroups: sci.math Keywords: Merten's Conjecture On Sat, 23 Jan 1999 "Nelson G. Rich" wrote: >What I meant to say was that I recall a while ago hearing about >a conjecture true for all n < some very large integer N, but >false for N (and possibly other integers as well). The point >being that even what appears to be a "reasonable" conjecture may >turn out to be false after all. Does anyone know any examples of >this sort of thing? The cannonical example may be Merten's conjecture. Recall that the Mobius function u(n) of a positive integer n is 0 if n is divisible by a square (greater than 1), and (-1)^k if n is a product of k distinct primes. If you let M(x) denote the sum of the Mobius functions of all the positive integers less than x, then it seems that M(x) never gets very large. This is important, because if can be shown that if there exists a constant C such that |M(x)| is always less than C sqrt(x), then the Riemann Hypothesis is true. Von Sterneck conjectured that |M(x)| is less than (1/2)sqrt(x). This is indeed true up to very large values of x. However, it fails for the first time at x = 7725038629, because the sum of the Mobius functions of all the integers up to this number is 43947. Merten's made the weaker conjecture that M(x) is always less than sqrt(x), so he was conjecturing C=1 instead of 1/2. However, Odlyzko and te Riele proved in 1985 that Merten's conjecture must fail for some x less than e^[3.21 x 10^64], although that bound can probably be improved. Anyway, this is an example of a conjecture that would be extremely important (implying the Riemann Hypothesis) if true, and that is empirically true farther than we could ever hope to compute, and yet it is ultimately false. ============================================================================== From: dik@cwi.nl (Dik T. Winter) Subject: Re: conjectures that are true for large n but are actually false Date: Mon, 25 Jan 1999 13:28:02 GMT Newsgroups: sci.math In article <78ftle$qvg$1@news1.rmi.net> Kurt Foster writes: > There is the "Mertens conjecture" that M(x) < sqrt(x) for x > 0, where > M(x) is the sum of the Mobius function mu(n) over n =< x. The first > counterexample was found in the early 1980's, and was a respectably large > number. The counterexample itself was not found (it was beyond the computational possibilities). It was proven that a counterexample exists in the neighborhood of exp(32097025772922655869740000186211307099797144540349062682805321651.697419). Nor was it shown that that would be the smallest counterexample. See A.M. Odlyzko & H.J.J. te Riele Disproof of the Mertens conjecture J. reine angew. Math. 357(1985)138-169 -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/