From: Bob Silverman
Subject: Re: Computations which could (in theory) disprove the Riemann Hypothesis
Date: Tue, 02 Jan 2001 19:55:48 GMT
Newsgroups: sci.math
Summary: Consequences of Riemann Hypothesis
In article <92t600$kpd$1@spiffy.ox.compsoc.net>,
rick@ox.compsoc.net (Richard Heylen) wrote:
> I am looking for a relatively easily comprehensible, true statement of the
> form "If the Riemann Hypothesis is true then there is no number with and such a property>."
Try the following.
If the Generalized Riemann Hypothesis is true then there is no
prime p whose smallest primitive root is greater than 2 log^2 p.
--
Bob Silverman
"You can lead a horse's ass to knowledge, but you can't make him think"
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==============================================================================
From: "Charles Matthews"
Subject: Re: Computations which could (in theory) disprove the Riemann Hypothesis
Date: Tue, 2 Jan 2001 21:05:48 -0000
Newsgroups: sci.math
Richard Heylen wrote
>I am looking for a relatively easily comprehensible, true statement of the
>form "If the Riemann Hypothesis is true then there is no number with and such a property>."
I pointed out about 15 years ago that a reformulation by Andre Weil of a
generalised Riemann Hypothesis (based on a theory about the "explicit
formulae" of prime number theory), as a statement of positive definiteness,
lent itself to computational tests. These are to do with the positive
definiteness of some finite matrices, which can be defined solely in terms
of the distribution of prime powers. This work was never published, but I
believe Richard Pinch once mentioned it in a paper.
In these days of plentiful computational power, it might be possible to do
something more with this approach than I attempted then. The matrices in
question can be expanded by adding further rows and columns - this naturally
makes it harder for them to be positive definite. By extracting some
quantitative measure of this you can get a decreasing sequence of real
numbers, which according to GRH tends to a non-negative limit. Intensive
computation might reveal some trend, for example a conjecture as to the
value of the limit (might actually be zero).
It seems rather unlikely that direct measures of the distribution of primes
would refute RH if they simply look at the first billion primes or so - I'm
not an expert in analytic number theory, but there are so many known zeroes
of the zeta-function on the line that the early primes must behave much as
predicted. GRH looks at distribution in APs also, and is less studied. I
suppose any computational tests ought to have some aim beyond just crunching
more numbers.
Charles
==============================================================================
From: "Dik T. Winter"
Subject: Re: Computations which could (in theory) disprove the Riemann Hypothesis
Date: Wed, 3 Jan 2001 02:46:12 GMT
Newsgroups: sci.math
In article <978469921.6444.0.nnrp-02.c2ded8d9@news.demon.co.uk> "Charles Matthews" writes:
> It seems rather unlikely that direct measures of the distribution of primes
> would refute RH if they simply look at the first billion primes or so - I'm
> not an expert in analytic number theory, but there are so many known zeroes
> of the zeta-function on the line that the early primes must behave much as
> predicted.
The last time I was connected with the search it was found that the first
1.5*10^9 zeroes were on the line. (The computations involved about 1.5
CPU months on a CDC Cyber. After that, or rather earlier, it became boring.)
From a private communication I understand it has been expanded to 2*10^9
zeroes or something like that. What appears to be a reasonable inference
from that search is that zeroes can get arbitrarily close together, but
just do not hit. I disremember the actual smallest separation found among
those zeroes, but it was on the order of 10^-14, or some magnitudes larger
or smaller.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
==============================================================================
From: "Hermann Kremer"
Subject: Re: Computations which could (in theory) disprove the Riemann Hypothesis
Date: Tue, 2 Jan 2001 23:33:35 +0100
Newsgroups: sci.math
Richard Heylen schrieb in Nachricht <92t600$kpd$1@spiffy.ox.compsoc.net>...
>I am looking for a relatively easily comprehensible, true statement of the
>form "If the Riemann Hypothesis is true then there is no number with and such a property>." The assumption of the proof of the Riemann
>Hypothesis should be necessary for the proof of the statement (at the
>current state of mathematical development). It should be computationally
>feasible to search a relatively large number of candidates looking for a
>candidate that has the property. The property should involve really
>simple-to-understand concepts and especially not complex numbers (i.e not
>involving the zeroes of the zeta function).
>
>I would like to use this as an example of a computation which (in theory)
>could settle the issue of the truth of an important hypothesis but which
>in reality is extremely unlikely to do so mainly because it would be
>"cheating". I think that mathematicians have an interesting meta-theory
>about the derivability of results which says that we will
>only establish the truth of the Riemann Hypothesis after we have spent a
>lot of time and effort discovering and exploring new areas of maths. For
>the Riemann Hypothesis to be false and for it to be discovered to be false
>by mere computation would be be a source of shock and outrage even though
>we cannot rule it out as a possibility at our current level of knowledge.
In the paper "On the representations of xy + yz + zx" by J. Borwein and
K-K. S. Choi (in DVI or PostScript)
http://www.cecm.sfu.ca/preprints/1998pp.html , preprint 98:119
it is shown that there are at most 19 integers which cannot be represented
as n = xy + yz + zx with integers x,y,z > 0, where the first 18 numbers can
very easily be computed and the 19-th one does only exist if Riemann's
Hypothesis is false.
Regards
Hermann
--
[deletia --djr]