From: cet1@cus.cam.ac.uk (Chris Thompson)
Newsgroups: sci.math
Subject: Mertens' Conjecture [was: Re: Goldbach conjecture]
Date: 13 Nov 1997 15:33:16 GMT

In article <34672962.D26@wing.rug.nl>,
Meinte Boersma  <m.boersma@wing.rug.nl> wrote:
[...]
>But "probably" isn't necessarily "likely" in this case.
>Take for example Mertens' Conjecture. It is a stronger
>version of the number theoretic reformulation of the
>Riemann Hypothesis.
>
>If I remember correctly the conjecture was proven wrong in the
>early 90's by Herman te Riele and someone else (his name slipped
>my mind), 

Andrew Odlyzko.

>          but the smallest counterexample was in the order of 10^63 !

Rather larger than that, if we are talking about finding an x such that
|M(x)| > x^{1/2} (where M(x) = \sum_{n<=x} \mu(n) ). Not that the work
in question actually generated such an x exactly. What they did was to
find values of y such that

    h_K(y) = \sum_{\rho} k(\gamma) exp{i\gamma y} / \rho\zeta'{\rho}

lay outside [-1,+1], for a suitable smoothing function k(.), to be compared 
with

    M(e^y) = \sum_{\rho} exp{i\gamma y} / \rho\zeta'{\rho} 
             + exponentially small terms

where the sums are over the zeros \rho = 1/2 + i\gamma of the Riemann
\zeta function. [I am ignoring the fine details of convergence in a 
thoroughly cavalier way here...] In particular they found

  h_K(-1404528968059299804679036163039978112740059199978973803996590762.521505)
     = +1.061545 approx
  h_K(+3209702577292265586974000018621130709979144540349062682805321651.697419)
     = -1.009749 approx

This is enough (by pre-existing theory) to show that M(x)x^{-1/2} achieves
values at least that large and small as x -> \infty. For positive y, as in
the second case, one can view this as a weighted average of values M(x)x^{-1/2}
for x around e^y, i.e. exp(3.2*10^63), not "order of 10^63"! For negative y
there doesn't seem to be such an intuitive way of looking at it.

Of course, this doesn't mean that there are not much smaller values of x such
that |M(x)| > x^{1/2} --- there undoubtedly are. The authors guesstimated
"we do not expect |M(x)x^{-1/2}| > 1 to occur for x < 10^20 and maybe not 
even for x < 10^30".

Chris Thompson
Email: cet1@cam.ac.uk